c.1600, probably an alteration of quease (c.1550), from Old English cwysan "to squeeze," of unknown origin, perhaps imitative (cf. German quetschen "to squeeze"). Slang expression to put the squeeze on (someone or something) "exert influence" is from 1711. Baseball squeeze play first recorded 1905. Main squeeze "most important person" is attested from 1896; meaning "one's sweetheart, lover" is attested by 1980.
main squeeze, put the squeeze on someone