Many details of the implementation are different in the individual states of the EU, since the health sector has so far not been harmonized. My experience is from Germany, where there exists a mixed system in which you are not allowed to have no health insurance. If you are self-employed or earn enough, you can switch to a private insurance company.

You brought up one very important thing when you wrote about monetary trouble and self-employment. Because even a mandatory insurance scheme as the one in Germany has holes in it. I recently learned that, if you are self-employed, have private health care and run into financial trouble, you can end up in a state of "inactive health insurance", where you still have basic health insurance, but you always have to go to the emergency ward at the hospital. So far, a few ten thousand people are affected by this (out of 80 Million), but it is a de facto hole in the system.

However, when I point this latter situation out to people around here they usually become quite enraged. In a way, this reinforces my point about the perception of health care as an ethics issue.