Part 1: I used to do these joke freestyle raps in Scottish accents that made my friends lose it. My first name is Joshua, so, once I started one with "my name is Josh, I don't eat squash."
Ever since people started calling me Josh the Squash, than just The Squash, than just Squash.
Part 2: I used to throw ragers at my house. My friend Chris started making .img's that advertised my parties to send to people via AIM or post on the local message board that were hilarious and featured me tweaking wasted and cats and other random shit. They always advertised the party as "Mosh with the Squash" or "Squashmosh".
Finally I was just squashmosh.