As a verb, squash means to beat, squeeze, press, or crush something into a flattened mass. As a noun it denotes the family of tendril-bearing plants with leathery rinds and edible fruit, while squash is also racket game played in a closed-walled court with a rubber ball.
Quash means (1) to set aside or annul by judicial action, and (2) to suppress forcibly and completely. When squash is used figuratively, its meaning can come very close to quash in the second sense.
Once the peel is soft—you should be able to squash it between your fingers —and the liquid has reduced, squeeze the muslin bag to extract the pectin. [Daily Mail]
Remove from the heat and once they’ve cooled down enough to handle, squash them one by one with the palm of your hand. [Books Live]
Egypt on Wednesday finally recovered its Internet access after the Mubarak regime forced a blackout in a failed attempt to quash dissent. [Electronista]
David Cameron is fighting a battle to quash a mounting insurgency by Tory MPs over Europe and relations with their Liberal Democrat coalition partners. [Tribune Magazine]