This perspective of public vs private interest is rather puzzling to me -- it's more a matter of short-term vs long-term thinking. In the long-term, it's in the private interest of those having sustainable companies to have a broad base of educated and healthy workers to employ. The short-term interest of any single firm has a different calculus, but all private companies would benefit from a stronger public education and health system.
The goal of government, in my opinion, is to focus on long-term objectives: ensuring a sustainable healthy and educated population.
It's similar case for education. It's not that I'm paying for someone else's kids school so much as I'm paying so that one day, when I need a smart educated nurse to take care of me... it'll likely happen at a reasonable price because there are so many educated people who have decided to be nurses.