If my house catches on fire, and someone doesn't put it out, there is a reasonable chance it will spread to other houses. So it's a societal issue - at least at the local scale. As a societal issue, it makes sense for each member to be forced to pay for it.
If I break my leg, there is no increased chance that my neighbor will break his leg. So it's a personal issue. My neighbor receives no benefit from my leg being repaired, and is not harmed if isn't repaired. So why should he be forced to pay for it?
If you want to say that public health care should exist to provide vaccinations, in order to prevent the spread of disease to others in society, that would be a reasonable argument.