But not everyone is (or becomes) healthy or unhealthy to the same degree. If I have no family history of health issues, or I simply want to take a chance that I'll remain healthy, why should anyone else tell me how much I should value my health?
That some people in society look at the circumstances of their birth and post-hoc reason that they don't want universal healthcare because it wouldn't have benefitted them is not the point really.
Incorrect, everyone gets sick to the same eventual degree: death. You will require medical care.
> or I simply want to take a chance that I'll remain healthy
No one remains healthy, period, everyone gets sick, everyone.
You could be the healthiest person in the world and get punched by someone who gives you an expensive disease through blood contact; you could randomly get cancer at any age no matter how healthy you think you are. So this notion that you think you should pay less is based on a naive understanding of how health actually works; you cannot predict the future. You could be disabled tomorrow through no fault of your own. As such, the system must be designed to consider that everyone will be sick at some point and most people will be healthy most of the time, everyone pays and the sick are covered, problem solved for everyone. Social medicine works better than any other approach, this is a fact proven around the world. Those objecting to it are doing so on ideological grounds, not on rational ones.