If I lived in some European country, the government would take money out of my paycheck to provide healthcare. Since I live in the U.S., my employer is effectively doing the same. In terms of value for money, there isn't much of a contest -- Europeans spend considerably less per capita on healthcare generally, and get outcomes that are generally better according to a wide variety of measures (expected lifespan, infant mortality, et cetera). But the Republican (and libertarian line) is that the American solution is better anyway, because government isn't coercing me into any particular arrangement. My employer is. Hey, wait a minute.
What's really going on in either case is that some organization is acting as an agent for providing healthcare services. Private solutions generally are claimed to be superior because people have a choice about what provider to use, and can choose the one they want. Government is supposed to be inferior because people don't have a choice, and have no control over what it does. But I have no choice at all about what my insurance arrangements are, and I do have at least a vote giving me (in combination with others) influence over the government.
With this in mind, let's take a look at Republican tax rhetoric. The line we've all heard thousands of times by now is that taxes ought to be reduced because "it's your money". But there was at one point in our history the notion that the government is the agent of the people, that tax money is our money, to be used for joint social purposes, like retirement support, education, and so forth, taking advantage of economies of scale which would be impossible if everyone had to make their own arrangements. If Republicans don't buy that, then they must not think that in our democracy, the government is the agent of the people. So what do they think it is?
But then again, Republicans are the people who passed a Medicare drug benefit in which the goverment (unlike all European governments) is literally forbidden by law from negotiating good prices from drug companies (as it does for Medicaid, according to laws written by Democrats). And who run a National Labor Relations Board which almost invariably rules against labor. So maybe they know exactly what government is... when it's run by Republicans.