Unlike some countries, the national health system in France requires that you pay into the system for a period of time before you are covered. Normally, this means you have to be employed in France in some capacity so that you're registered in the "cotisation" system (which includes health coverage, retirement and a couple other benefits). You can't pay into just the health system, you have to contribute to the whole shebang.
Depending on how your company is set up, there are a couple of ways you can manage this.
You could establish a French branch and pay yourself a salary through this branch. The branch would have to register as some form of business entity, report results and the whole deal. The set-up process is a royal pain, but once it's up and running, it's reasonably easy to maintain.
You could establish yourself as an "independent" and then bill the company in the US for your services (assuming you are still active in your company) performed while you are in France. You would have to register with the RSI under one of the applicable statutes and pay your taxes and cotisations based on the results of your business here in France.
Or, if you're not currently active in your business back in the US, you pay yourself dividends and will have to purchase private health coverage through an insurance company or broker.
Cheers,
Bev