To say that Hayek had “no problem” with a national health care program is rather misleading.
Yes, Hayek believed in a minimal safety net. But he also believed that anytime the government starts providing these “services” for “free” they will get much worse, unless there are outside competitive forces which will keep it somewhat honest. And he certainly would not have wanted to keep expanding and expanding any program until it uses up all of our wealth and still gives worse and worse service...which is where the US is headed.
Hayek would have advocated, I think it is fair to say, for example in the UK, sort of allowing the NHS to continue to exist, but don’t expand it, and make competitive forces work for the rest of the population that want to avoid the death trap of the NHS religion.