I expect that a conversation with such a person would go like:

  Person: No, free.
  You: So how will the doctor get paid?
  Person: By the government.
  You: Where does the government get its money?
  Person: Oh...

People usually have the knowledge, they just haven't made the connections.

There are 7 billion people on Earth right now. You are going to find some that don't know what the hell "breathing" means if you just try hard enough. Using this to pick apart at the word "everyone" is just as pedantic as picking at the specific definition of "free" in that question.

Even more, in most systems I know the amount paid for health insurance is fairly explicit. Where I live you actually get a bill from a non-state company (some things may be subsidized from other taxes, and if you can't afford it you also get a subsidy, though) so no, people don't think it's "really-really" free. It's "people with more money contribute more".