So, the problem boils down to this:
There is a 60 year old man named Bob. He's worked at a construction site every day he's had a job, which is most of his life since 16. He works hard, because he has two sons and a daughter he wants to have a good education. He budgets so that the family goes without a lot of things--him most of all. He's been recently diagnosed with cancer, and his life savings aren't enough to foot the bill and send his daughter to college. Without college, she'll just be looking at waitressing at a dead-end restaurant for the rest of her life. He stays up late, cutting more and more from the budget, and it comes down to that choice.
There is a 20 year old man named Jimmy. Jimmy's out of college, out of work, and out enjoying life. By "enjoying life" I mean "smoking pot and doing stupid dangerous things." He's recently been cut off from his parents' money, and is living off of his life savings, which are substantial, thanks to generous family members, but are rapidly dwindling. One day, while drunk and high, he jumps off of a moving van and breaks his leg. He doesn't have enough money to fix the leg and continue living like this for another month, so he's trying to bum some money off of his relatives.
90% of people would give money to Bob. 90% of people wouldn't give money to Jimmy. The difference between supporters and opposers of UHC is who they see when they picture the people who will benefit from this. That, and economic reasoning, which is generally in the backseat of these stories.