What do you mean by philosophical justification? There are a number of ethical frameworks that would like that more or less.
But it's certainly useful to have it. First of all even "free health care" is not actually without cost and no sane person would claim that. The idea of an insurance is just that you spread the damage on a large group of people, rather than letting the individual receive the full impact of an illness that is likely more accidental than within their personal responsibility. The idea being that when spread on a larger group of people the impact per person stays manageably small, while for a single individual it could be devastating. So it's the calculation of a planable inconvenience vs an erratic event that, in the worst case, can destroy the individual in question and leave the group 1 short.
What the "free" refers to is that you have no or very little costs at the time of using the service. Which is actually a very good idea for several reasons. First of all in the event of a sudden injury you're not at your best, so you might be physically and psychologically impaired (shock aso), so the less you have to worry in that situation the better. Also usually the sooner and more professional an injury is treated the better, as illnesses and injuries often tend to get worse if untreated, while treated or even prevented medical problems might only cause a small inconvenience or be reversed completely.
From it's general idea, it's as far as I can see, not in an obvious conflict with most ethical frameworks and it can be a net positive for both the individual in need of it and for society who profits from what that individual is able to contribute because of the treatment. And that's not just physical labor, but also experience, compassion and whatnot. Not to mention that usually human rights would also demand that you help each other if that's possible. Or rather it asserts the value of their rights to an equal and dignified existence which makes that more important than a slight inconvenience on your end.
In the end it depends on what morals and what ethical framework you subscribe to and how it's implemented. Like the assertion that you'd need to coerce medical personal to perform their service is usually completely unfounded. Helping people is usually a job that is highly demanded, highly valued and lots of people volunteer for that. That being said these people have to be cared for as well so it's not without cost and that cost is to be taken care of either by those who provide it (unlikely), by those who use the service (possible for smaller stuff, impossible for larger operations) or by society as a whole (meaning people have regular expenses even if they are not sick themselves).
And there's a spectrum on what you would like to and are able to cover as "preventable medical problems" and how vast the expenses for the individual ought to be and how they should be distributed. And there are probably people who claim to be an isolated self and thus see this as an intrusion upon themselves. Now for most people that is simply not practical because you are not an isolated self but on a daily basis make use of the inventions and the work of other people and even the purely transactional perspective doesn't really work as you're likely also making use of what people gave you "for free" under the assumption of it being a society. So most people neither are nor what to isolate themselves to that extend or are even able to do so.
Or the more likely people complain that in the implementation of it, they'd pay more than they take out or more than other people and would be better off handling their healthcare alone. Well if you pay more than you take out, you're rather lucky cause that means you're healthy and don't have to struggle with medical problems, but that's not true for all people and it might also just be a fluke for you. And that rich people are supposed to pay more is more or less a consequence of the distribution of wealth in society. Like if everybody had roughly the same then the burden for health care would be roughly the same but if you demand a contribution to healthcare that is above your wage or above what you need to survive than healthcare would simply not be that high on your priority list. But that's not a general problem but rather a consequence of a different socio-economic problem that just spills over.
Edit: An obviously with infectious diseases and especially within a pandemic protecting others by provided access to medical treatment and vaccines is also doing yourself a favor because otherwise the virus is keep coming back and has most likely way more negative consequences (lockdowns, isolation, death, aso.)
