Yes. I'm an immigrant and have access to free health care.
Also, as an EU citizen you can get a European health insurance card which entitles you to state medical care in 32 European countries. Not all these countries have free medical care but, in any case, you'll get the same level of treatment for the same price as a local.
I've stayed out of this conversation until now as I don't have a vote but this seemed like an apt point to jump in. Before jumping out again I'll just add that if I did have one it'd be (for countless and ever-increasing reasons) a big fat 'yes'.