If an Orthodox Christian wears their baptismal cross to a private school, do you think it'd be okay for that school to tell them that they can't wear it?
I don't think it is okay, and it doesn't matter if it is private because it is still infringing on that persons freedom to practice their religion. That's also why I believe private schools affiliated with religion can't and shouldn't force those not of their religion or sect to behave, believe or dress a certain way. Sure, they can choose staff and students with religion in mind, but if they hire someone or accept a student who practiced a different faith, they can't interfere with their rights.
There are religious affiliated schools who require you to attend services. If its Christian and you're a Muslim, then why should they force you?
A school should have the right to refuse someone. But once that person is within the schools walls, the school must respect that persons freedom to practice their religion.
If you choose to attend that school, it isn't by force you are doing anything, it's by your choice when you attend. That is the difference between private and public schools and why public schools are required to remain neutral, because the student does not have a choice in attending. Private schools can do whatever they want because you can always choose to leave. By attending that school, you are stating you respect the schools policies, religion, dress code, etc., and give up any right to complain.
For example, if I attend a private university (Protestant), I must comply with their requirements, even if I disagree with them. My option is to attend a different university. Very simple.
Also, I am not saying I either agree or disagree with the schools decision, only I agree the choice is with the school, no one else.