Not everything is so black and white. People describe themselves as "conservatives", even though they don't want to bring back slavery and take away women's right to vote. People describe themselves as "left-wing", even though they don't want the government to nationalize every business in the country.
Pretty much every developed country in the world has realized that it's good for health care to be widely accessible. This generally means one of two approaches (or a combination):
Given a choice between these two alternatives, it's entirely understandable that someone who calls themselves a libertarian might prefer the second one. It's a much smaller burden on employers. It means that it's easier for people to be self-employed, start their own small businesses, etc. It also means that companies have more flexibility with their labor pool, and it means that companies don't have to keep paying for the health care of employees who retire or who are laid off.
The same argument applies for a universal basic income instead of a minimum wage, or for government pensions instead of private plans. In all of these cases, the practical choice is between the government writing checks to people, or the government making rules that people must follow. I can definitely understand why some people with libertarian leanings would choose the former.
Consider also that, in the absence of any government policy on health care, you may end up with a monopolistic/oligopolistic health sector. Big business has all the downsides of big government, except that you don't even get a token vote. And you can't really choose not to get health care in the way that you can choose not to shop at a certain retailer. It's possible to design a health care system where all services are provided by private entities, and the government just writes checks. In such a system, you actually end up with a lot more choice than you would with a few national HMOs.
Consider also that Alaska, arguably the most libertarian-leaning state in the US (possibly tied with New Hampshire), earns 68% of its revenue from taxes on oil companies, and sends every resident a check for about $1200/year. You'd be hard-pressed to find anyone who wants to get rid of that.
Consider also that the Liberal Democrats, the closest that Britain has to a libertarian political party, support the NHS just as strongly as they support civil rights.