The way I see it, the government is OK with taxing me and using that money to protect me and my goods from harm when my house catches on fire, and to protect my neighbors from the same, this is the fire department. The government is likewise willing to tax me and use the money to protect me from assailants who would be looking to do me harm or to unlawfully take my goods, this is what the police department does. The government can also ta me and use that money to protect me and my fellow citizens from the actions of foreign nations, this is the purview of our military. The government can also tax me and use that money to maintain and regulate the roads I travel on. Having accepted these facts as true, I would jump to the broader generalization that the government has the right to tax me and use that tax money for my own protection. Allowing for the fact that there are numerous different ways I could come to harm, and it is impractical for the government to protect to attempt to protect me from every possible bad thing that could happen to me, we should consider the ways I am most likely to come to harm over the course of my life. If we accept police and fire protection as a baseline for protections that it is right for the government to provide, then we need to ask, if a person is more likely to suffer some form of life threatening disease then it is for them to be threatened by crime or fire. I believe that health risks pose a greater danger to me than do those presented by fire or crime or foreign invasion and so by that reasoning I believe people do have a right to healthcare. I also believe that while providing healthcare to everyone is expensive, since I am more likely to benefit from the existence of universal healthcare, the return on investment is also greater.
Obviously there are other arguments to be made in favor of how providing healthcare to everyone helps to protect those people who could afford healthcare from diseases passed on from those who don't have healthcare, the benefits of preventive care vs receiving treatment after the fact, and the much vaguer question of what is our duty to our fellow man.
TL;DR. If the government can protect me from dying when my house catching on fire or when I get mugged, then it has the right to protect me from cancer and heart attacks too.