As far as I know, Xavier (and Dowling, Assumption etc. though I’m less familiar with them) doesn’t turn down a prospective student because they are not Catholic, or are not Christian, or are openly atheist or whatever. They charge them higher tuition, make them agree to take a theology class and go cash the checks.
I’m saying that might change if those schools are swamped with families wanting in now that Kim is going to foot the bill. It would not be surprising if some private schools are soon faced with far more demand than whatever their target enrollment number is, and how they decide who to take and who to turn away is going to be interesting.
My guess is religion may play a role, but rack rate tuition might also go up significantly such that there’s still a good chunk of out of pocket money that needs to be spent even after voucher money.