Then Czech Republic must be an exception in the EU - pretty much every other country that I lived in worked on the simple principle of "you are a citizen/resident = you are entitled to care". Even if you didn't make any payments ever(due to lack of a job, or being homeless or disabled for example) you would still be 100% covered.
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.
Really? As long as you are properly employed in any EU country you should also be covered in that country. There are some issues regarding cross-border, meaning a German is working in Spain and needs medical attention in Germany that would be invoiced to the Spanish insurance.
In Germany free-lancers face issue, they need to get their insurance somewhere. But thay have to get, there is no such thing as uninsured German resident.
It is not an exception, it's almost the same at least in Slovakia, Poland, Hungary and Austria. It is not true that you'd be always covered - you'd be covered only if you're registered in the system and entitled to be a part of it (by the virtue of being a resident); if you have your residence outside the EU, you're entitled to nothing; if you have your place of residence in other EU country, you're entitled only to treatment of immediately life threating issues - you still need travel insurance to travel to other EU countries if you want e.g. dentist covered, also you can't injure yourself doing sports and such, that would not be covered at all by the basic insurance.
It's true that you'd be covered even if you never paid but you have to be entitled to be a part of the system - and thus you're insured. Also the money for treatment will be later demanded from you, on top of insurance payments and interest (personal experience); of course there are options for the socially less able (the government pays insurance for them, that's why it might seem free) but as a software engineer, no it's not true.
If you read my comment properly you'd know that I'm talking about uninsured people.
EU citizens are able to move to another EU country are eligible for the same benefits as locals as long as they are working and paying taxes there and for X time after that (I can't remember the exact number).
In the Nordic countries that's extended quite a bit for citizens of other Nordic countries (even if the other country isn't in the EU), and I wouldn't be surprised if there are other groups of countries in the EU with their own similar agreements.
None of the above says anything about the level of welfare the individual EU countries have, however. If an EU member has a below-average welfare system then you might end up with nothing simply because you don't meet the requirements for receiving benefits.