Jérôme,
I'm glad to hear you were able to localize the issue. We know of many, many users that use the Language Switcher plugin without any conflicts with JFBConnect. However, there have been cases where problems have cropped up. We've never been able to recreate the problem on our own development server and the users with the problem were ok either switching off the language switcher or doing things differently (I don't have the information on what they chose to do).
If you want us to investigate further, we gladly will. To do so, can you please Private Message super admin credentials to us so we can see exactly how you have things configured. Then, we can try to determine what specifically is the cause.
You can PM me by using the mail icon under my name to the left.
Thanks,
Alex