There's no way for government to protect your property rights without infringement on my negative rights to my liberty either.
This is particularly obvious in the case of private ownership of land. Without private property rights I would be free to walk across and otherwise make use of that land it is your property rights which take that freedom away from me. And this isn't a theoretical issue either: prior to the enclosure of land in the 15th-19th centuries (https://en.wikipedia.org/wiki/Enclosure), most land here in the UK was common land that anybody had a right to make use of to graze animals and make a living. The expansion of private property directly took away that right and deprived many of their livelihoods. Even today we still have the Right To Roam (https://en.wikipedia.org/wiki/Freedom_to_roam#United_Kingdom) and at least walk on private land which provides for significantly more freedom than is available in the US (there is close to zero danger of being shot just for walking somewhere in the UK).
For other property it's less obvious, but enforcing your property rights still requires encroaching on my liberty. For example, without property rights, I would be free to walk into a shop and take whatever I wanted to. Or rock up to your unlocked car and drive off with it. The only way that this is prevented is through active enforcement by the state. Ultimately, if I persist in doing these things then I will be physically imprisoned. And it takes significant resources to run the policing and justice systems, just like healthcare. Now obviously nobody thinks I should just be able to walk up and take your care. But other forms of private property are controversial. In particular the idea that you ought to be able to keep all of that wealth that people give you (without taxation). You are only able to hold onto that wealth in the first place because the state enforces your right to keep it. There is nothing a priori that says that the state (which ultimately represents its citizens) must always enforce that right.
To summarise, many "negative rights", including the right to private property, have exactly the same ethical problems as "positive rights". At which point it becomes a simple matter of what you consider moral and important. You're right that state-provided healthcare deprives you of wealth that you could otherwise have. But the reverse is also true. You having the wealth deprives others of healthcare that they could otherwise have. You just happen to think that property rights are more "natural" and important.