While the missionaries had theirs paid by their employers, my guess is that NBC is doing the same.
As far as I know, all of these clinics are private businesses, and a part of the larger hospital.
I haven’t found anything yet on who is paying for NBC.
UNMC here in Omaha is the Nebraska Medical Center so it is public. Emory, UNMC and the others have their isolation/treatment centers sponsored by the Fed which is why they have to undergo periodic inspections and certifications by the CDC. The Texas hospital is not a government sponsored center as far as I know.