Absolutely. If you suddenly became sick or injured, you'd be treated, first of all. You don't have to produce the cash upfront or be turned away from the hospital.
Then there would be a lot of alternatives, a lot of possible ways this expense might be paid, including Medicaid and other local, state and federal benefits, charities, donations. Your friends might do fund raisers to cover your medical bills. You might hit the lottery. Hospitals and medical facilities are pretty good about working with people for a great long time, because they are used to having broke people confronted with massive bills. If you did qualify for Medicaid, ran up a substantial bill, and then died, the state might come back and get your home. But you wouldn't be wanting to leave your descendents stuff with your legitimate bills unpaid anyway, would you?
But why are you worrying about such things right at the present? Be working on finding another job soon. Campaign for health care reforms. But don't sit around and work yourself into a lather trying to figure out what MIGHT happen to you! How many people do you know that this has happened to?