"to crush," 1560s, from Old French esquasser "to crush," from Vulgar Latin *exquassare, from Latin ex- "out" (see ex-) + quassare "to shatter" (see quash "to crush"). Related: Squashed; squashing. The racket game is first recorded by that name in 1886, originally it was the name of the soft rubber ball used in it.
"gourd fruit," 1640s, shortened borrowing from Narraganset (Algonquian) askutasquash, literally "the green things that may be eaten raw," from askut "green, raw" + asquash "eaten," in which the -ash is a plural affix (cf. succotash).