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.
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.