Liao and Kurita attempt to offer government healthcare to all subjects, though what gets delivered varies. (For example, the backward planet of Annapolis in the Combine - described in A Time of War - can only provide "traditional medicine practitioners" to most villages, and the average lifespan is about 50 years for commoners. Nobles get better healthcare and often live twice as long.)
Marik and Davion have policies varying by planet. You'll find options ranging from "what's medicine?" on Davion Outback worlds to "inexpensive mandatory private insurance" on El Dorado (again, covered in ATOW), with all variations in between.
The Lyran system is described in the House Steiner Handbook, p. 122. Generally, high quality medical care is available - if you can pay for it.
"The use of advanced techniques and the need to travel to select
facilities [in other star systems] where they are available for treatment often dramatically increases
the costs of such procedures. This fact tragically underscores
the class distinctions between wealthy Lyrans and their poorer
countrymen. Though medical insurance companies and special
“personal emergency” loans available through some hospitals do
exist, their coverage is often less than comprehensive and the costs
of their premiums or interest rates can often put a patient in debt for
years—even decades—after the fact."
So Solaris VII is likely to have private healthcare systems, and MechWarriors in Solaris City will likely have very high premiums.
