In fact, although as a French resident for more than 15 years I could not vote, until I saw that the referendum was apparently turning on an anti-immigration wave of sentiment I would have voted to leave. This was because I feel that the UK has always been unwilling to form an integral part of an increasingly centralised "European Super State" - "no" to the €uro; no to Schengen; no to the immigration policy; no ..; no....; no ....
So in my view, the referendum result to leave was the right result for the wrong reason -xenophobia, anti-immigration, etc
As regards UK citizens living in France and using the (wonderful) French health system, I don't think that we will see any differences at all.
The UK wants to free itself of the European bureaucracy - we are still part of Europe!!