There are workers employed by the government - federal, state, and local (a quick google tells me that NYC owns 11 hospitals 6 diagnostic centers and 80 clinics, for example, there seems to also be a state owned hospital here and I believe there is a federally owned VA one as well). There are workers employed by the for profit private sector . There are workers employed by the non profit private sector. In most cases hospitals in all three of those categories treat both people covered by  public provided insurance and those with private insurance, as well as those with no insurance at all (emergency rooms are required to provide urgent care regardless of the patient's ability to pay for it).  

So there isn't a yes or no answer to your question - is a doctor in private practice whose patients mostly rely on Medicare 'employed' by the government? What about a doctor working in a public sector hospital whose patients are primarily those with private insurance?