Private healthcare is still perfectly legal and fine, as is private health insurance, if you deem the state services are not suitable for you.
Hospitals should be helping people, not ripping them off >.<
What about the police department? Or the fire department? Or the post office?
Any government is, at its core, socialist. Rather than everyone funding their own private army, the state socialises the cost and provides national defence.
For other parts of society, it may also make sense to have everyone pay in to provide service which is to the public good. Police, fire, postal services are canonical examples.
In most countries, health care is also socialised. As is garbage collection, libraries, and road building.
The question - for most reasonable people - is where do you draw the line? Buses are socialised - what about taxis? Doctors are socialised - what about Lawyers, or accountants?