I'm not sure which private schools your are talking about or which government money. I am only familiar with the funding of the school in my parish and that my kids attend. We do have some state links for vouchers and such. But there is no money that we are dependant on from the federal government. If the Department of Education went poof! tomorrow, we wouldn't notice.
They rely overwhelmingly on tax deductible donations.
Lose the tax deductions, most lose most of their funding.