Paying for the care of others is also the insurance model, which is not socialism. I think where you lose support from part of the population is when the system is being abused. I understand France for instance has an abnormally high number of depressions, as it is a common way for a lazy employee to take some time off at the expense of the system. In the US I understand there is an abnormally high number of people on disability benefits.

The hostility I think comes not from the fact that it is mandatory, but from the fact that if it is managed by the state, it will be badly managed, abused by a minority, and you end up over-paying while getting a bad service yourself the day you need it.