Oh yeah I forgot the legal drinking age in USA is 21

(Also, the way you're phrasing your argument makes it sound like people get lazy the moment they're healthy. Like they won't work so that they can afford food, rent or other living (and fun) commodities. It’s not like that. Many people who are poor, or those who have an average income, often work harder to have the money and time to do the things they want. If they’re sick, how could they do that?)
Sure, many people are neglecting their health, but guess what could change that? Proper education. A doctor or nurse explaining what will happen to their bodies. Check-ups (nothing like a good doctor-shamin to get you healthier!). And medicine (In case their obesity and general unhealthy lifestyle is caused by genes.). And, well. Some people are just lazy. Some people /would/ exploit that system, but there are so many more hardworking people who would benefit from it.
No, this is not knocking the doors of communism and socialism. It’s utilizing the nation’s funds to aid the bigger picture.

I see that the thing that doesn't seem to sit right with you is that there is more money taken from the rich. But their finances will not take as big of a blow as you seem to be estimating. Again, this is a couple of percent of their salary. (Hell, with how big USA is, less than a percent of everyone’s salaries would be sufficent to give people basic healthcare needs.) They will receive just as good healthcare as everyone else, if they would need it.
Now, I propose that instead of unnecessarily building the military, some of that money could go into healthcare needs as well. Keeping both future soldiers and civilians healthy is important for the nation.

Sweden also have a progressive tax system, and free healthcare is working just fine. The only things that cost money is the medicine, and that's because different medication companies can have different prices that aren't covered by the state's finances. Sweden is, as of today, one of the best countries to live in because of this, and they're doing all this with help only from Finland (And that help is not born of necessity, but of alliance. Healthcare doesn't cost too much to the state, especially when it's not at war.).
A lot of the world’s economies have progressive taxing, and while the way you’re thinking is admirable, an economy where everyone pay equally wouldn’t survive in the world today, and it also would not be fair to those with less money. Then your argument would be turned around and become what it is today; Why should they pay so much when they have so little?

But... I'm 18...

They would pay an equal percentage of their salary to the government. Sure, if you have more money, the percentage will be a bigger sum, but it probably won't matter that much in the long run since you have money to spare. The opposite goes for the opposite; If you have a small salary and pay your percentage, you will pay less money than someone who has more than you, but just enough of your salary that it won't be too damaging to your finances.
If the American health system worked like that, americans would be a lot healthier, regardless of financial status.

Except those are puppies, and they can in no way be shitty, but poor mother

I can still relate to feeling uncomfortable when you're full of other people's shit

Oh yeah, I'll just
*picks up brains and places them back inside the skulls they came from*
There! All better!

That is not what they are saying. Free healthcare means that everyone will pay for everyone's healthcare, instead of some people having to put themselves in debt because they got sick or hurt while others have a shitload of money laying around.
(And this is a different matter entirely from free healthcare, so you bringing this up is pretty random, but whatever.) This woman is praised for caring more for her dog than a really expensive wordly possesion. It's nice to be reminded that there's more than greed and hate in the world, and some people wouldn't put this much money into their dog. There is, how ever, people like this, and moments like this. It makes us happy to see it.

Gintama

call it q then