The official requirements of the Archdiocese of Philadelphia say that an applicant must be a "practicing Catholic" and provide a letter from his pastor.
I would guess that it varies. When I was in Junior High, at a Catholic school, we had one teacher who was Protestant. (She didn't teach religion.)
We had teachers who were admitted atheists. At one point, we even doubted that our freshman year religion teacher was Christian (she would tell Old Testament stories and laugh afterward...I think she was just a very liberal Catholic, though).
I WISH we had Orthodox teachers, so apply away!
As others said, I think it depends on the school. Perhaps you have to be Catholic INO (In name only) at the very least, or if they're desperate, they will just make you attend Masses and teach other subjects. And in my area, Catholic schools do NOT pay well with their limited budgets, so I think that getting a good, pious Orthodox teacher would be the least of their problems.