DENVER, Pa. -- Jesse Martin, an Old Order Mennonite farmer here, lives much like his 17th-century Dutch ancestors. He shuns electricity and cars. He believes in self-sufficiency, so he opposes insurance and government aid.

But nine of his 11 children suffer from serious diseases. Desperate to save them, he has tapped into 21st-century high-tech medical care. In the process, he is posing tough questions for the U.S. health-care system and for nonprofit hospitals, which are expected to provide charity care in exchange for tax...