I keep coming back to Hayek on this - because government-sponsored health insurance is not government-run healthcare. Maybe it's my British roots that make this so clear to me (if the British Tories proposed a universal health insurance scheme, with care provided by private doctors, nurses and hospitals, it would rightly be regarded as a major shift to the right.) Here's Hayek's discussion of why this is not heresy for libertarians and conservatives of the old school:
Nor is there any reason why the state should not assist the individuals in providing for those common hazards of life against which, because of their uncertainty, few individuals can make adequate provision.
Where, as in the case of sickness and accident, neither the desire to avoid such calamities nor the efforts to overcome their consequences are as a rule weakened by the provision of assistance - where, in short, we deal with genuinely insurable risks - the case for the state's helping to organize a comprehensive system of social insurance is very strong... Wherever communal action can mitigate disasters against which the individual can neither attempt to guard himself nor make the provision for the consequences, such communal action should undoubtedly be taken," - The Road To Serfdom (Chapter 9).
Demanding and helping people insure themselves in a context where emergency care is already guaranteed is not socialism. It's prudence.