Nowhere in the Constitution does it say health, healthcare or healthcare insurance are rights.  Healthcare is a service, thus it is a privilege.  

And even if you think it is a right, government does not give rights so government shouldn't pay for it either.  

Let's look at it this way.  Our Bill of Rights specifically says I have a right to keep and bear arms.  Applying your logic that the government pays for it, then they should pay for my guns, ammo and accessories - agreed?