If this is the case for you, then breathe easy. Playing a little of the other racket sports, such as squash, tennis, or badminton, isn't going to hurt your child's ping-pong prowess. The human brain is pretty smart when it comes to telling the difference between different sports and the strokes used. In fact, anything that makes your child have to watch and hit a moving ball is only going to help his or her coordination.
The only proviso I would make is to suggest that your kids don't play a lot other racket sports while learning their basic table tennis stroke technique, which will take maybe 2-4 months, depending on the individual. Once the basic patterns are grooved in, let them play what they want. They'll get the benefits of extra exercise, as well as a bit of a break from ping-pong every now and then.
Actually, one more point I'll mention is to limit the amount of other sports prior to a competition. This is mainly to avoid the risk of injury while your child is performing a sport that he or she doesn't play that often. Different sports use different muscles, so it is easy to strain little used muscles while playing another sport.
