I can’t speak to Mr. Clinton’s specific current health insurance plan, but I’m sure he’s covered under his wife’s insurance. COBRA would’ve expired in July, 2002, so it can’t be that. I’m also reasonably certain that, as an ex-president, he’s still eligible for care through the military health care system, although if he has insurance, that will still be billed.
All federal employees, including members of Congress, the President, and the Supreme Court justices are eligible for insurance through the Federal Employees Health Benefit program. The local mail carrier is eligible for the same insurance as the President, in fact.
That said, health insurance is covered under the Health Insurance Portability and Accountability Act, which governs privacy for all health information, including claims.