Ooh, my specialty (I'm on the Princeton squash team).
The answer is yes and no.
Yes, because squash is a racket sport, and the skills that are developed in other racket sports (hand-eye coordination, etc.) are often easy to transfer.
No (and this is the larger part), because squash is entirely different from tennis and badminton. Firstly, the dominant spin in squash is backspin (i.e. every shot is effectively hit with slice, and topspin is basically useless). Secondly, squash is played on a court similar to a racquetball court. Also, the movement is very different - much more lunging, and a different movement dynamic than either tennis or badminton.
Read Wikipedia for more information:
Squash (sport) - Wikipedia, the free encyclopedia.
Now, all of this said, if you start playing squash you will likely have a shorter learning curve than someone who had not played any racket sports beforehand, but the learning curve will still be tough. For most collegiate teams, though, it won't be as easy as taking your other racket skills and transferring them to squash.
If you think you're going to want to try out for a college team, or play extensively, it would be a good idea to start now (assuming you have access to a court and can buy a racket, goggles, etc.).
Telling the coach you'd like to walk on is fine, assuming you also tell him that our only experience (extensive as it may be) is with other racket sports.
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