The phrase to squeeze someone until the pips squeak and variants mean to exact the maximum payment which a person can afford.

It was coined by the British Conservative politician Eric Campbell Geddes (1875-1937), who, while campaigning for re-election as MP for Cambridge, Cambridgeshire, in December 1918, referred to the reparations exacted from Germany following the First World War; the Cambridge Daily News of Wednesday 11^th contains the following report:

SIR ERIC GEDDES.
Germany to be Squeezed “Till the Pips Squeak.”

Sir Eric Geddes followed up his big meeting at the Guildhall on Monday night by addressing another crowded assembly in the large hall at the Beaconsfield Club on Tuesday night. He had another enthusiastic reception […]

“Squeezed Till the Pips Squeak.”

Dealing with the question of indemnities, Sif Eric said: The Germans, if this Government is returned, are going to pay every penny; they are going be squeezed as a lemon is squeezed—until the pips squeak. My only doubt is not whether we can squeeze hard enough, but whether there is enough juice. (Laughter.) It is right that they should pay. They brought this war on; they asked for it, and they have got it. And now let them pay the bill for the goods they have asked for. (Applause.)

In a letter published by The Times (London) of Friday 15^th December 1933, Eric Geddes went back over the circumstances in which he had coined the phrase:

“UNTIL THE PIPS SQUEAK”
GERMAN INDEMNITIES IN 1918
AN ELECTION PHRASE EXPLAINED
TO THE EDITOR OF THE TIMES

Sir,—Fifteen years ago, on December 11, 1918, to be exact, I made a speech in the Beaconsfield Club Hall, Cambridge, during the course of a campaign in what became known as the “Khaki Election.” Language tended to be heated and exuberant in that election, the restraints of the Great War being over and the economic facts which mellow and subdue as yet unforeseen in any strength and clearness. In that speech I used a phrase which caught the fancy of the moment by giving expression no doubt to feelings and expectations which ran riot in the public mind, unaware of the disillusionment and the wisdom that the years were soon to bring. The phrase was this: “squeezing Germany like a lemon until the pips squeak.” It has attained unmerited longevity despite my hope that it might have died and been buried with other things born in a passionate election, which though only 15 years ago has seemed at times to be a century past.

It is often said that the British Labour Party politician Dennis Healey (1917-2015) used the phrase when referring to plans to increase taxation for high earners; for example, The Illustrated London News of May 1978 described Healey as beginning,

in his first Budget, in March, 1974, by putting into practice his desire “to squeeze the rich until the pips squeak” by increasing taxes on the well-off and by changing from indirect to direct taxation.

This attribution seems to rest solely on the following passage from Mr Healey promises action on profits of food manufacturers and retailers, by Christopher Thomas, published in The Times (London) of Tuesday 19^th February 1974:

Lincoln, Feb 18
Mr Healey, shadow Chancellor, focused the election campaign sharply on to inflation tonight, accusing the Conservatives of destroying the fabric of society and introducing class feeling.
Promising to “squeeze property speculators until the pips squeak,” he said that Lord Carrington, Secretary of State for Energy, had made £10m profit from selling agricultural land at prices 30 to 60 times as high as it would command as farming land.

However, Dennis Healey always denied having used the phrase, whether as Shadow Chancellor of the Exchequer, from 1972 to 1974, or as Chancellor of the Exchequer, from 1974 to 1979. The Times, which is an anti-Labour newspaper, may have misquoted him. Already, on Monday 12^th November 1973, it had published Leadership for Mr Wilson at a stroke, in which David Wood wrote:

It is at least plausible to assume that the ordinary voter, including large numbers of historic Labour voters, simply find Mr Wilson* and the Labour Party unbelievable and unworthy of trust. No wonder.
[…]
In Government, they frankly said that taxing the rich and the better-off would not alone solve the problem of economic and social inequality; in Opposition, forgetting the rate of inflation, they would tax the upper working class until the pips squeak.

* Harold Wilson (1916-95), British Labour statesman, Prime Minister 1964-70 and 1974-76).