Well, here's one answer. The league is a public service, supported by the taxpayers. The parents of kids in private schools pay taxes, too. By the same reasoning that home-schooled kids can play on public school teams- because of the word public and its meaning- private schools should be able to participate in the Public league.
Now, the terms of that participation should be open to debate. Some have proposed a 1.5 multipler for attendance at private schools, for example. They clearly have competitive advantages in some sports; probably not in others.
