As you correctly point out, it is of the same kind as the required auto liability insurance, although it goes further. As the fee of the mandatory public health insurance is (for the same services) proportional to revenue, it is ALSO a kind of tax, in that it imposes some kind of solidarity (the rich pay a bit for the poor), while the car insurance is flat fee of course (well, not depending on your wealth, but rather on your objective risk factor).
But in both cases (actually more so for the health care) it is a relatively SMALL fee which can cover a potentially LARGE expense, and which, as your example points out, nobody can be sure not to need one day. In other words, to avoid a small expense, people who would refuse a minimum coverage do something objectively stupid. In any case, it is a small (relatively to your wealth) fee, it is not going to avoid you buy a bigger house, or go on a fancier holiday or anything, and you run a serious risk that one day, you have a SERIOUS problem. That's why it is objectively stupid: the gain you get from refusing it doesn't weight in with the potential danger you put yourself into.
As to the nanny state, I'm also against law enforcement that *obliges* you to BEHAVE in a safe way (as long as you're not increasing the risk for others of course). Nobody can determine how much pleasure you get from smoking, and whether or not you are willing to sacrifice your health for the pleasure of smoking, say. So although for most people, this is a stupid action, as you cannot quantify the "gain" (pleasure in doing so), you cannot objectively show that it is stupid.
But the insurance fee is really small, so you can quantify it. The gain isn't big, it is equal to the fee.