In Sweden every visit to a doctor or hospital is billed, as is every night spend at the hospital, so they are definitely equipped to handle it.
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.
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'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.
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.
The only case that would maybe trigger a payment would be if you were a visitor to EU and had a baby - but most likely no hospital would bill you as hospitals are just not equipped for billing, they don't have the staff or the equipment to do such weird things as issue bills for treatment to people. I know at least several non-EU people who had treatment in EU and no one ever asked them for any insurance info to bill them, unless it's something major and very very expensive there's just no point, it would most likely cost the hospital more to figure out how much and how to bill you in the first place.