In most European countries, the right to healthcare insurance is linked with working thus paying through mandatory contributions from one's payroll to support this system. Citizenship is not an issue, but being a legal resident in the country is.
Then, at least in France, it has been demonstrated that having illegal immigrants in good health condition was better for the society than having illegal immigrants in bad health condition. As we don't let them die (because of human values, not religious ones), we end up providing some healthcare. And healthcare costs less if it is applied before a condition becomes serious. SO we end up having a minimal level of healthcare provided to everyone legal or not. Of course, our current rightist government is trying to get back on this.