They are partly right. On the other hand, one US state has the size and population of one european country.

Different systems affect at the choices available to population. Here we also have conditioned choice. For example, I have both public and private health insurance. The first one is mandatory, you pay for it in your taxes and you always have it. The second is optional. Here, people who has private health insurance usually use both. It is common that you do test and check outs at private healthcare and then go with those to the public. In fact, private refer you to the public healthcare for serious things like cancer. On the other hand, people pay private hospitals to have nicer rooms. So you do have conditioned choice here too.

Republicans are right about being an state issue in the sense that if you can freely travel among states, and one state has public healthcare and others don't, that many people would travel to get "free healthcare". Or pressure their representatives to implement the same system other state has. You have named Florida as an example of better healthcare plan. If I'm not mistaken, many old people (who would need better healthcare) retire there. So there you have an example of how a system influences how people behave.

So in a way, healthcare is a state issue with the potential to influence demographics.