The difference being that in the Netherlands, we all have to carry ID, immigrant or not. I don't really understand what the big deal is, I always have my ID card in my wallet. I've never been asked for ID anyway, not even at bars, unlike in the US. It's like you trust bartenders more than police officers. If that's the case, you have bigger problems than this new law.
Actually, you can have private health insurance companies and still have insurance for everyone. By regulating things a bit more, and using a risk pool. This is how it's done here in the Netherlands:
http://en.wikipedia.org/wiki/Health_care_in_The_Netherlands