The problem is there's a false scarcity. Healthcare is expensive for people in the US right now, so the idea of paying for someone else's healthcare via universal healthcare doesn't sound like a good thing, it sounds like taking the money you need for your healthcare to pay for someone else's.
It's NOT like that, but people have it in their heads that it is. And it's very hard to change a made up mind when there are people rallying against it to protect their own private interests.