Your first choice for a game of Squash is the Lyttelton Recreation Centre at 25 Winchester Street, Lyttelton - actually it's the only choice if you want to play squash in Lyttelton.
You can book a court online at Pay2Play and get instant access, there are often a few racquets and balls left around in the mezzanine overlooking the courts, but I recommend bringing your own equipment if you can.