If you’re self employed and you live and work in France then your business has to be registered in France, as a French business. The logic is that unlike an employee for a company, where you have two separate entities - the company and the employee - a self employed person is the business, ie they are the same entity. Therefore if you live in France, so does your business. It makes no difference where your clients are.
You then pay social security contributions based on your earned income, which entitles you to healthcare plus other benefits as appropriate.
In order to register a business in France (micro entreprise might be your best business structure) you need to satisfy the requirements as regards residence, right to work etc. I believe that as a rule US citizens on a long stay visa are allowed to register as micro entrepreneurs in France, but the decision is made at préfecture level and interpretations sometimes vary from one préfecture to another. Since this is critical for you, I think the best plan would be to contact the préfecture of the place you’re planning to move to and check with them that they don’t envisage any problems. Ideally get something from them in writing.
I also believe that some US citizens do live here temporarily and continue to earn money from a US based business with the authorities turning a blind eye, but technically that is against all the rules and I don’t think it would wash if you’re looking at a permanent move. If you’re here for the long term you need to join the French social security system and pay your dues here.
Hope this helps.