So just like with car insurance, that minimal level of coverage would only include health care costs incurred by others as a result of your actions? The minimal required car insurance isn't there for you, it's there to ensure that others can recover their costs from you in the event of an accident. Unless you're paying extra for comprehensive coverage, the insurance company won't pay out for your own treatment or the repair of your own vehicle.
The only thing that might come close in the domain of healthcare would be mandatory coverage against the possibility of accidentally transmitting a contagious disease. It wouldn't cover the cost of your treatment, but rather the cost of treating others whom you accidentally infect. The premiums would be minimal (so long as one is vaccinated) and it wouldn't look anything like traditional health insurance.
Of course, even the "minimum" level of car insurance is only mandatory in connection with driving on public roads. Putting aside the minor issues of a power imbalance sufficient to imply duress at the best of times and the fact that you'll be paying taxes for those roads whether or not you're allowed to use them, this is somewhat analogous to terms and conditions for use of someone else's property. What you're proposing amounts to mandatory insurance just for being alive—essentially on the basis that denying you healthcare you can't or won't pay for might make others feel bad—which is a rather different proposition.
Really, though, a better question is what penalty you propose for not buying whatever "minimal level of coverage" you think people should have. My suggestion is simple and doesn't require any special intervention: If you're not insured, can't pay for it yourself, and can't convince anyone else to pay for it for you as a voluntary act of charity, you don't receive treatment. That is the natural punishment for not buying health insurance. There is no need for any other.