Wasn't there a writeup recently that wealth distribution isn't materially distinguishable from luck? [1] Your argument that the healthy/rich subsidize the poor/sick is tantamount to saying the unlucky should fend for themselves. That's an easy argument to make if you're lucky. Your luck may turn at any moment, and then if it does, you're SOL.
Further, if the moral argument falls flat for you, the self-centered argument is that if someone is too poor to get their horrifyingly contagious disease treated until its too late and it spreads to you, you're still sick, and you may still die, cover or no cover. It's why the fire department is socialized -- if the fire spreads to your house, it still burns does it not?
Especially as you argue you're not particularly well off necessary, you should be the strongest advocate for yourself, and therefore, for socialized medicine.
The world is moving there -- at least towards a two-tier system -- and this argument is on the wrong side of history.
[1] https://www.technologyreview.com/s/610395/if-youre-so-smart-...
[EDIT] I also wanted to throw in there the self-centered argument for socialized higher education. If you believe in the value of a college education, then you believe it will increase the productivity of that individual. A more productive individual makes your country more competitive, boosts GDP, moves the markets higher relative to that of other countries. This in turn benefits the wealthy who likely hold equities. Speculative, of course, but the argument seems consistent to me.