I found the solution without knowing what the problem was. I searched into the troubleshooting section, followed all the steps and I am back to where I was originally. Very Happy because I have meanwhile tried 3 other browsers and I do not like any, even equally to Firefox!

In your More System Details on right it appears as if you have a blank UserAgent so that could be a reason why Facebook is saying your web browser is not supported.

https://support.mozilla.org/en-US/kb/how-reset-default-user-agent-firefox

James - I followed the link you gave me (after I had things back to normal however) and I did not find the entries that should have been found... I guess that by then the expected culprits were not there anymore. Thank you for answering.