There might be a weenie nuance here. I'd mash a potato, reducing it to a nice smooth creamy texture. But I'd crush a garlic clove, leaving it mainly intact but split open. I'd squash a tin can before I put it in the recycling, leaving it flattened out.
I agree with Winklepicker. The choice of verb depends on the form of the food before and after the action. All three of these verbs, plus the less "official" moosh, mush, smoosh, smush, squish, etc., are essentially onomatopoeic.