There is no free health care in France for anyone. French people or residented non French people pay 8% of their earning towards health care which is a part if their income tax deductions.
The arrangement between the two countries mentioned in the above posting only applies to people who are over 65 + their spouse of any age. It used to apply to everyone until about a year or so ago when it was changed with only 2 weeks notice.
British people under 65 receive only basic health care in France and other EU countries obtainable under the European Health Insurance Card system which replaced the old E 111 forms they had to get. (details of this can be obtained on the internet) No longer can British people under 65 get free health care in France for on -going chronic illness like they used to.
We seem to be straying off the original topic here about commuting crew and their income tax arrangements. I doubt very much if any commuting crew are over 65 at the moment, but who knows in the future perhaps there might be some.