These are some of the things I would consider when deciding if something should be run and/or managed by the government.
1. It is something that everybody needs.
For both defense and health care this is obvious.
2. It is something that need to be organized on a vast scale.
Obviously true in both cases.
3. It is not something that can be run well for profit.
More arguable, but experience shows in both cases that maximizing profit involves reducing quality of service. For mercenaries this means dragging out conflicts while minimizing actual combat. For medicine, it means forcing people to use cheaper medicine even if better ones are available and denying coverage as much as possible.
4. It is something that is better run by governments then others.
For military service this has been historical true, countries that depend too much on mercenaries end up in trouble. It's more arguable for medicine, but the experience of European countries says that at the least it can be run as well by the government as by buisness.
5. It's in the government's best interest that this be handled well.
Obvious for the military, more arguable for medicine. But I think it's in a countries long term interest to have healthy citizens.
Jay