What are eligible jobs? In most cases, eligibility is based on whether you work for an eligible employer. Your job is eligible if you:
are employed by any nonprofit, tax-exempt 501(c)(3) organization;
are employed by the federal government, a state government, local government, or tribal government (this includes the military and public schools and colleges); or
serve in a full-time AmeriCorps or Peace Corps position.
If you don't meet these criteria, the Department of Education's regulations create a two-part test of other circumstances under which you may still be eligible:
(1) your employer is not "a business organized for profit, a labor union, a partisan political organization, or an organization engaged in religious activities, unless the qualifying activities are unrelated to religious instruction, worship services, or any form of proselytizing;"
and,
(2) your employer provides any of the following public services: emergency management; military service; public safety; law enforcement; public interest law services; early childhood education; public service for individuals with disabilities and the elderly; public health; public education; public library services; and school library or other school-based services.

I think they make it pretty clear private schools don't count.