The term eager beaver denotes a keen and enthusiastic person who works very hard.

Aided by the phonetic similarity of the two words that compose it, eager beaver alludes to the animal’s industriousness, the beaver being remarkable for its skill in constructing huts of mud and wood for its habitation, and dams for preserving its supply of water—cf. the earlier phrase to work like a beaver, meaning to work hard and productively.

The term eager beaver originated in—or was popularised by [see note]—US forces’ slang during the Second World War. The earliest instance that I have found is from Slang of the Yank, consisting of “dizzy definitions used around army camps, naval and coast guard stations, air corps and marine bases, and sent in by boys in the armed forces themselves”, published in The Sheboygan Press (Sheboygan, Wisconsin) of 23^rd September 1942:

Eager Beaver—Soldier imbued with the desire to please his superiors with a show of exuberance for unpleasant tasks which his buddies look upon with distaste.

According to the subsequent instances of eager beaver, the term was peculiar to the US Air Force; the following is from The Salt Lake Tribune (Salt Lake City, Utah) of 12^th December 1942:

‘Bombardier’ Will Furnish New Words to Language

By Ernest Foster
Hollywood (UP)—Are you riding on the shack? Or having a dry run?
Whether you’re a bubble chaser or a bird dog, a shack rat or an H.B., remember it takes an eager beaver to please a clank-clank.
Double talk? Not at all. It’s the newest slang—bombardier talk.
It’s only about two years old. That’s the age of that most colorful, cocky and swaggering of today’s brood of Uncle Sam’s eagles. They’re the boys who drop the block-busters from such a height that those on the ground can’t even see their huge Flying Fortresses.
[…]
The color of these new glamor boys of the air and their slanguage figures in “Bombardier,” RKO Radio’s air air [sic] saga starring Pat O’Brien, Randolph Scott and Anne Shirley.
Since the picture was one of five suggested for filming by the war department and was made chiefly on location at Kirtland field bombardier school and other army bases, everything in it is authentic. Even the picturesque slang.
“Riding on the shack” means you are doing all right, whether with a girl or the Norden bombsight. “Having a dry run” means getting nowhere, either with a girl or hitting a target.
A “run” is the straight flight used for bomb dropping. A “shack” in reference to bombing means a bulls-eye.
But a “shack rat” is superlative for woman chaser and a “run” can also mean pitching woo. A “bubble chaser” is a milder term for girl chaser, but an “eager beaver” does not mean that at all. It refers to an alert, efficient student cadet.
Dealing with a military secret is “I’ll put you on with the outside knob.” That refers to the final adjustment on the bombsight before releasing the bombs and is slang for “I’ll knock the hell out of you.”
“Lie detector” is the photograph which shows how bombs land in reference to the target. “Meat house” or “sweat box” is the bombardier’s compartment on a plane. “Ring shiner” is the eye circles a bombardier supposedly gets from his bombsight.
To a student, an instructor is a “hen,” a class is a “brood.” To an instructor a student is a “gadget.”
If you’re an “HB” you’re good—that means “hot bombardier.” But if you’re “gig happy” you have too many demerits and may “wash out”—be eliminated. If you’re “paunchy” you’re a sloppy, lazy cadet.
A bombardier doesn’t “bail out” or “hit the silk.” He “takes a walk,” and reminds himself to put on his parachute on entering a plane by saying “I don’t want to walk without you, baby.”
A parachute is a “jump sack.” A “parachute hike” refers to a training or disciplinary march wearing a parachute.

On 18^th December 1942, The State Journal (Lansing, Michigan) gave a more detailed definition:

“EAGER BEAVERS” are those aviation cadets in pre-flight school who most thoroughly apply themselves to tasks in ground school and on the drill fields.
Here their “eagerness” will determine whether they make the grade in pre-flight school, go on to primary, basic, and advanced schools, and finally emerge as combat aircrew pilots on one of America’s battlefronts.
Hence a familiar greeting to under-classmen from upper-classmen is: “Are you eager, mister?”
It remained for Aviation Cadet Beaver A. Kinsel of San Antonio to confound all the other “eager beavers.”
He is known as Eager Beaver Beaver.

The fact of being an “eager beaver” was formally rewarded in the US Air Force; on 9^th January 1943, the Lincolnshire Standard (Boston, Lincolnshire, England) reported that a Mr H. Appleby, of Boston, had received

a cablegram from his son, Cadet Charles John Appleby, R.A.F., who is in America, saying that he has gained his “wings” and has been granted a commission.
In a recent letter to his father he said that he was training with Ray White, of Manchester […]. John said that Ray won the “Eager Beaver” prize for being the best cadet in the class while undergoing basic training.

Note—I have found a single instance of eager beaver from 1942 indicating that the term was apparently not confined to military circles and was used in a more general sense; it is from the commercial advertisements and public announcements published in the Santa Ana Register (Santa Ana, California) on 30^th November (the same notice also appeared on 1^st, 2^nd and 3^rd December):

If you’re an “eager beaver”, you’ll hurry right down to Goodwill for used merchandise. 417 W. 4^th.