It's definitely a good idea to buy squash shoes. Ordinary trainers are OK to get you started, but they aren't really designed for the sharp turns and hard acceleration required in squash. Tennis shoes have a very different structure and aren't really suitable at all. You can use badminton or other “indoor court” shoes, but, as with every other sport, you'll always get the best performance by having equipment that’s specific to your sport. Additionally, some clubs insist that your footwear have soles that won’t mark the court surface.