How does one oppose the principle of access to healthcare being universal per se?
That would probably be opposition to being forced to pay the cost of another's self inflicted sickness/injury.
I have a hard time seeing how cancers of the skin (which are on the rise), and other kinds, etc., as well as high blood pressure, hyper tension, kidney stones, and so on are self-inflicted. It's vastly hereditary.
So people with good genes should subsidize those with bad genes? More relevantly, people with good health habits should subsidize those with bad habits?