The idea is that you are always insured for essentials, so the question is to find out under whom at that time. No idea what specifically determines the insurer

The idea is that you are always insured for essentials, so the question is to find out under whom at that time. No idea what specifically determines the insurer for someone entering the country. But coverage will begin back on entry and they will request membership payments for the entire time, even if you did not "use" that coverage. As for refugees specifically, I'd expect that to be mostly a question of whom they sign on with first. The public insurance has the same rate regardless of provider, so it wouldn't make much of a difference, and governmental support will probably pay that bill anyway.