It was re-used by Denis Healey, Chancellor of the Exchequer, on presenting budget to Parliament in May 1978.

(Said of German war reparations following World War I by Sir Eric Geddes (1875-1937), British Conservative politician, in December 1918 at the Guildhall, Cambridge, England.) My only doubt is not whether we can squeeze hard enough, but whether there is enough juice. I will squeeze her until you can hear the pips squeak.

We will get everything out of her that you can squeeze out of a lemon and a bit more.

Its origin, though often attributed to Churchill, is actually from Sir Eric Campbell-Geddes, who served as First Lord of the Admiralty (civilian head of the Navy) in 1917-19.

Curiously, the epithet “pipsqueak,” meaning a weak and/or insignificant person, seems to have no real connection to “make the pips squeak.” Although it appeared about the same time (1910), “pipsqueak” employs “pip” in the sense of “something very small” and “squeak” in the sense of “small, weak sound” to convey the sense of a young child or powerless adult who can only squeak in protest.

Pipsqueak appears to have no relation to the above saying:

The term first appeared in print in reference to the heavy reparations demanded from Germany after World War I (“Dealing with the question of indemnities, Sir Eric said: The Germans, if this Government is returned, are going to pay every penny they are going to be squeezed as a lemon is squeezed - until the pips squeak.” 1918). The sense is that if a fruit is squeezed very strongly the pips would shoot out, perhaps at least figuratively, making a “squeaking” sound as they fly across the room.

To “squeeze (someone) until the pips squeak,” meaning to exert heavy pressure on someone in order to extract money, information or simply obedience, is definitely a reference to the “pips” in a fruit. As suggested by the Word Detective pips refers to fruits seeds:

Download pip and squeak food