The attributive phrases chicken-and-(the-)egg and hen-and-(the-)egg are used of a situation in which it is difficult to determine which of two entities or events should be considered the cause and which the effect when each appears to depend on the prior existence of the other.

Those phrases refer to the traditional problem of which came first, the chicken/hen (to lay the egg) or the egg (to produce the chicken/hen).

This problem is first recorded in Quaestiones Convivales, by the Greek biographer and philosopher Plutarch (circa 46-circa 120). The following is from The Philosophy, commonly called, the Morals written by the Learned Philosopher Plutarch of Chæronea. Translated out of Greek into English, and conferred with the Latine Translations and the French, by Philemon Holland, Doctor of Physick (London: Printed by S. G. for J. Kirton […], 1657), by the English scholar, physician and translator Philemon Holland (1552-1637):

THE THIRD QUESTION.
Whether was before, The hen or the egge?

[…] When I was one day at a feast which Sossius Senecio made unto us, […] Alexander the Epicurean […] tooke occasion hereupon, to set on foot that doubtfull question of the egge and the bird, which had busied and amuzed the heads so much of great naturallists, and searchers into the causes of naturall works, and namely to know, whether of the twain was before; Whereat Sylla our familiar friend said: That with this little question of the hen and the egge, as with a small lever, screw, or such like engine, we shaked the great frame and weighty fabricke of the generation of the whole world, and therfore willed him to surcease and proceed no farther, to speak thereof. But when Alexander laughed at it, and made no more reckoning of it, than of a ridiculous question of no importance nor consequence at all depending thereof; my sonne in law Firmus began in this wise: I must here borrow (quoth he) the indivisible elements of Epicurus, and make use of those motes or atomy of his; for if it be true which he supposeth and laieth for a ground: That small principles should afford beginning to great bodies; it soundeth by all likelihood to great reason, that the egge was before the hen: for as far forth as by our senses we are able to judge, it is more simple, whereas the hen is a body mixt and compounded; and to speake in generality, the principle or element is ever first: the seed is a principle, and the egge full of seed, and less than the chick or living creature that is hatched of it: for like as the progess [sic] and prcoeeding [sic] unto vertue is of a middle nature, between the first disposition and the finall habit and perfection thereof; even so it should seem, that the egge is a certaine progress and advancement forward of nature, tending to make a living creature of the seed disposed thereto: moreover, as in a beast or such a living creature it is commonly said and received that the arteries and veines be formed first; semblably, good reason there is to hold that the egge was before the bird, as the continent before the thing conteined within: for so it is with very arts which make the first draught of their works grosly without forme and fashion; but afterwards give distinct figure and shape to every part therof, according to that which Polycletus the famous imager was wont to say: That their workemanship in poetry [= pottery?] was then most difficult and hard, when the clay and the finger naile met together: that is to say, when the worke was at the point to be finished: and therefore it standeth well to good reason, that the matter yeelding and obeying but slowly unto nature at the beginning, when she moveth and frameth by little and little produceth at the first, rude lumps and masses, not as yet brought into shape and fashion, such as eggs be; but as the same grow to receive the impression of some forme there is afterwards wrought out and framed a living creature within: for like as there is ingendered first a grub, which in time growing hard by reason of driness cleaveth and openeth in the end, and putteth forth another little winged flie, which we call Nympha, before it is a perfect bee; after the same manner, the egge here is the first subsistent matter of generation; for necessary it is, that in every change and transmutation, that must precede and have a beginning first which is to be altered & turned into another: see you not how cankers & caterpillers are bred in trees, and wormes in wood, either by the putrefaction, or concoction of humidity? and will any man deny that the said moisture went before; and that by order of nature, that which ingendereth is more ancient than that which is ingendered? for as Plato saith: The matter in all things that breed, serveth in stead of mother or nurse; and that is to be counted the matter, whereof the thing is composed and consisteth which is bred. And now for that which remaineth (quoth he, and therewith he laughed) I will sing unto those that be skilfull and of understanding, one holy and sacred sentence, taken out of the deepe secrets of Orpheus, which not only importeth thus much, that the egge was before the hen, but also attributeth and adjudgeth unto it, the right of eldership and priority of all things in the world: as for the rest, let them remain unspoken of in silence (as Herodotus saith) for that they be exceeding divine and mysticall; this only will I speak by the way: That the world conteining as it doth, so many sorts and sundry kinds of living creatures, there is not in manner one I dare well say, exempt from being ingendered of an egge, for the egge bringeth forth birds and foules that flie; fishes an infinite number that swim; land creatures, as lizards; such as live both on land and water, as crokodiles; those that be two footed, as the bird; such as are footless, as the serpent; and last of all, them which have many feet, as the unwinged locust. Not without great reason therefore is it consecrated to the sacred ceremonies and mysteries of Bacchus, as representing that nature which produceth and comprehendeth in it selfe all things.
When Firmus had discoursed in this wise, Senecio opposed himselfe and said: That the last similitude and comparison which he brought, was that, which first and principally made against him: For you mark not O Firmus (quoth he) how ere you were aware, you opened the world like a gate, as the proverb saith, even upon your selfe; for that the world was before all other things, as being most perfect, and reason would, that whatsoever is perfect, should precede the unperfect; the entire and sound go before that which is wanting and defective; and the whole before the part, for that there can be no parcell, but the whole thereof went before: for no man useth to speak thus: The seeds-man, or the eggs hen; but contrariwise we say: The mans seed, and the hens egge, as if both generative seed and egge did succeed and follow them, taking their own generation in them first, and afterwards paying again (as it were a debt unto nature) a successive generation from them: for need they have of that which is proper and familiar unto them, and thereupon are endued with a naturall desire and inclination, to produce such another thing as that was from whence they came: and hereupon it is, that seed is thus defined, to be a geniture or thing bred, having need and desire of new generation. Now there is nothing that either standeth in need or hath an appetite to that which is not, or hath no being: and we may plainly see, that eggs have their totall essence and substance, from that compact knot & composition which is gathered within the body of a living creature, & faileth herein only, that it hath not such organs, instruments, and vessels as they have; which is the reason that you shall never finde written in any history that an egge was engendered immediately of the earth; for even the poets themselves do say: That the egge out of which sprang Castor and Pollux, fell from heaven; whereas the earth even at this day produceth many compleat and perfect creatures; as for example, mice in Aegypt, and in many other places, serpents, froggs, and grashoppers, by reason that the principle and puissance generative, is infused and inserted into it from without. In Sicily during the time of the Servile war, much carnage there was & a great quantity of bloud shed and spilt upon the earth, many dead bodies corrupted and putrified above the ground, lying unburied; by occasion whereof, an infinite number of locusts were engendered, which being spred over the face of the whole island, spoiled and destroyed all the corne in the countrey: all these creatures therefore are bred and fed of the earth; and of their nourishment they yeeld a generall superfluity, apt to ingender the same kind, and that is called, seed; and for to be discharged thereof, by means of a certain mutuall pleasure, the male and the female match and couple together; and so some according to their nature, breed and lay eggs; others bring forth young ones alive; whereby it is evidently seen, that the primitive generation came first and immediatly from the earth, but afterwards, by a certain conjunction of [one] with another, in a second sort, they breed their young. In summe, to say that the egge was before the hen, is as much as if the matrice were before the woman; for looke what relation there is between the said matrice and the egge, the semblable hath the egge unto the chicken that is ingendered and hatched within it. So that, to demand how birds were made when there were eggs is all one, as to aske how men and women were created, before the naturall parts and generall members of the one sex and the other were made? And verily the members for the most part, have their subsistence and being together with the whole; but the powers and faculties come after those members; the functions succeed the faculties, and consequently, the effects or complements follow upon the said functions and operation: now the accomplished work or perfection of that generative faculty in the naturall parts is the seed or the egge: so that we must of necessity confess, that they be, after the generation of the whole. Consider moreover, that, as it is not possible that there should be concoction of meats or any nourishment, before the living creatur be fully made and compleat, no more can there be any seed or egge; for that both the one and the other, is made by certain concoctions and alterations: neither is it seen, how before the full perfection of a living creature, there should be any thing that hath the nature of the superfluity or excrement of nutrition; and yet I must needs say, that naturall seed otherwise, in some sort, may go for the principle and beginning of life; whereas the egge in no proportion answereth to such a principle, for that it hath not a subsistence first, nor any reason or nature of the whole, because it is imperfect. And hereupon it is, that we never say, that a living creature had any being or subsistence, without an elementary beginning: but we affirm, that there was a principle of generation, to wit, the power or faculty generative, by which the matter was transmuted, and wherein there was imprinted a generall temperature; and that the egge afterwards, is as it were a certaine supergeneration, much like unto the bloud and milk of a living creature, after nourishment and concoction: for never shall you see an egge engendred of mud; for that an egge hath a generation and concretion within the body only of a living creature; whereas there be an innumerable sort of creatures procreated and bred of mud and within mud. And to seeke no further for allegation of other examples to prove this, there be taken every day an infinite number of eeles, and yet never saw any man one eele, either milter or spawner, or that had any row in it. And more than that, if one let out all the water forth out of the poole, and cleanse it from all mud and mire, yet after the water is returned thither again into the place, there will be eeles soone ingendred. And therefore we may conclude necessarily, that whatsoever in generation hath need of another, can not chuse but be after it; and that which otherwise may be of it selfe, and without the other, must of necessity precede and go before in generation: for this is that priority whereof I speak. To prove this, mark how birds do build and make their nests before they lay eggs; women also provide cradles, clouts, beds, and swadling-clothes for their little babes, before they cry out, or be delivered; and yet you will not say (I trow) that either the nest was before the egge, or the swadling-cloth before the infant. For (as Plato saith) the earth doth not imitate a woman, but a woman the earth; and consequently, all other femals. And very like it is, that the first procreation out of the earth, was performed intire, and accomplished by the absolute vertue and perfection of the Creator, without need of such instruments, vessels, or secondines, which nature deviseth now and frameth in parents, by reason of their imbecillity and weakness.

The earliest occurrences that I have found of the attributive phrases chicken-and-(the-)egg and hen-and-(the-)egg are from letters published in the Boston Investigator (Boston, Massachusetts, USA):

– Of Wednesday 21^st March 1855:

To Dr. G. A. Hammett.

Dear Sir:—A press of business has prevented me from giving my immediate attention to the answers you have had the kindness to make to the questions I had propounded as a preliminary to a refutation of your proposition contained in the Investigator of November 1^st, 1854. I know proceed to perform the deferred task.
In the palmy days of attic philosophy, there arose a question which caused much debate in Athens. The question was:—“Which was first—the chicken or the egg?”
The answer of the Priesthood was: “The Chicken;” for, said they, God created all the fishes of the sea, the beasts of the field, the birds of the air, and man to rule over them all. Besides, (continued they,) the egg could not be laid or hatched without a chicken to do the act. “A past succession having no beginning is impossible;” and since there must have been a beginning, “the adaptation of the egg to produce the chicken does not indicate that it was formed by a pre-existing egg, therefore the succession began by the chicken.”
The answer of the Pythagoreans was: “The Egg;” for, said they, all chickens proceed from eggs—all things proceed from an original germ which is vivified by the heat and moisture of the elements. “Matter once existed in a state of chaos, and yet possessed at the same time certain inherent non-mechanical principles of order and harmony from which all present arrangements are derived.”
The answer of the Platonists was: “Both;” for, said they, the chicken and the egg are one,—the chicken is in the egg and the egg is in the chicken from all eternity. They contended that the existence of the chicken and the egg at any time was proof that both of them was eternal, as “nothing exists but one eternal now and one infinite here.”
The answer of the Peripatetics was: “Each;” for, said they, nature presents a succession of revolutions, births, and deaths, one preceding the other eternally, and it is impossible to find a beginning or an end to the alternation. “Each change must have occurred at a certain FINITE distance of time ago, but the whole succession is INFINITE.”
The answer of the Sophists was: “Neither;” for, said they, all the arguments used are equally false and equally true at the same time. They used against each party the reasons adduced by all the others; and they contended that as it could not be demonstrated that it was the egg, it must be the chicken, as that was the only alternative left; and that as it could not be demonstrated that it was the chicken, it must be the egg, as that was the only alternative left; and that as the two positions were respectively controverted, neither was true.
Diogenes, the cynic, laughed at them all, and would come forth, now with an egg, now with a hen, and presenting one or the other to the disputants, he would sneeringly ask, whence this egg? From a hen, was the necessary answer. Whence this hen? From an egg, was the unavoidable reply.
Your arguments, Dr. Hammett, might provoke similar questions and debates; for you have by your reasoning brought about the inquiry: which was first, the intelligent or unintelligent cause? You have proclaimed a decision a priori. You declare the unintelligent cause to be the true first cause, and the reasons you give for this solution might be used in the chicken and egg controversy with precisely the same force and effect as in this discussion about the pre-existence of an unintelligent or intelligent cause. A solution a priori, one way or the other, is perfectly impossible.
Like the hen and egg argument, yours can be reversed, and made to serve the purpose of its own refutation.

– Of Wednesday 18^th November 1857:

Nature’s Truths—“X. B.”

Mr. Editor:—One would suppose that persons engaged in searching out Nature’s truths, would find nothing in the harmony she teaches to engender animosities; and it amazes me, that because I simply expose some of “X. B.’s” egregious blunders, he should become so enraged. […]
[…]
It is a well known fact, that when the swamps in our western country are drained for a number of years and then burned, willows and aspens will spring up where they were unknown before. The same is true where the forests are cleared away, new kinds of timber frequently spring up; and when new soil is brought from a depth below the surface, new kinds of grasses and weeds will be produced. The circumstances are such that it would be almost impossible for any seeds to be deposited in such places; and the query arises, how do they originate? “All acorns come from oaks, and all oaks come from acorns,” like the chicken and egg problem quoted by “X. B.,” yet oaks will spring up where there are no acorns, and it is only where they come out of the ground, that their origin is any more obscure than a chicken originating spontaneously.