It means: beat the Miami Dolphins. However, after Dave Lundy published the novel "Squish the Fish”, it took on a perverted life of its own. In the book, Scooter thinks it means getting laid and pounding pussy.
After the Bills squish the fish and beat the Dolphins, I'm gonna celebrate at our tailgate party and squish my girlfriend's fish!
P1; dude me and Liz went on a Squish Date the other night.
P2; I didn't know you Two were dating
P1; no dude, a squish is someone who you love closely but in a platonic way
P2; oh you mean she's just your best friend?
P1; yep and we go on Squish dates because it's perfectly normal to have female best friends