Not the case for Poland either. If one is not employed and not registered as unemployed (authorities are doing all they can to keep the official unemployment statistics low), one has to proactively subscribe to "voluntary health insurance". In case one doesn't - pronto! one has no healthcare insurance. Of course, the constitution guarantees the healthcare, free-universal-healthcare blah blah - good luck suing everyone or anyone while being in such life situation.

Slightly different, but also extreme, case - Germany. There, one always is insured, only for the months one doesn't pay a debt is accumulating (you thought healthcare can incur debt only in US?). It's quite difficult to get out of German healthcare insurance actually, even while being EU citizen and moving out of Germany.

In fact, for a EU citizen who takes advantage of free movement of workers and relocates often between EU countries, one can fall pretty badly on its face healthcare insurance-wise.