Even if you don't have health insurance, in most of Europe you are fine. Especially in Central/Eastern Europe, where doctors in public hospitals work extra hours privately to make a better living.
Well, I think in all of the EU, you will be treated no matter what in an emergency. It doesn't matter if you have health insurance or not. That's the whole point of universal healthcare, something I've come to regard as a basic right in a civilized world.