Healthcare is no different from roads, schools, libraries etc.: everyone pays for other people using it, because otherwise you would have a crappy system. Taxing everyone to provide basic necessary services and infrastructure is the definition of society. Removing those taxes is the same as taking steps back to being uncivilised.
I am not joking when I say the US has one of the most expensive (per capita) healthcare system in the world, while also one of the worst. I mean, really, Cuba scores better on most healthcare metrics! They're not exactly known for being a wealthy people, yet they can somehow all afford better healthcare than the average American. They even sent about twice as many health professionals (in absolute numbers) to aid West Africa during the Ebola crisis than what the US did. Twice as many, from a population less than that of Ohio alone.
The US could spend less than it was spending and still have free care for all, and still have private insurance for those who want it.
There's a reason why a lot of Americans want to repeal Obamacare: turns out, they don't want to pay for other people's "free" healthcare.