They are under the impression that what'll happen is that their paycheck will go down by the cost of universal healthcare per person because the costs are taken out via taxes. Then they hear that some people will get it who don't pay taxes and they get indignant that they're paying and someone else is getting. Then they think about the difficulties they have with our current system, and picture putting something like the DMV in front of it, since that's what a lot of people have as their biggest reference for what the government does.
That's all because someone has a vested interest in making sure they understand it wrong, and no one is going to make a lot of money off universal healthcare so there isn't the same degree of motivation to teach people a more accurate understanding.
"Against" has billions of dollars and hundreds of thousands of jobs, and "for" has "human decency", "efficiency" and "why are we doing this to ourselves".
People hear that your paycheck gets a bit bigger, you go to the doctor when you feel sick, bills are mainly to keep you from going to the doctor for free aspirin and are lower than your copay, and you just ... don't deal with the billing anymore and think that sounds unrealistic. Entirely missing that other countries have done it, that the government already has a medical billing system, and dealing with paperwork is something the government does even better than "moving stuff from one place to another".