Zsolt,
Very sorry for the delayed response. I'm not sure if you're still having this issue, but simply changing the file type to UTF8 shouldn't have affected the language behavior. Joomla will try to load the language file based on the language set for your site (hu-HU) followed by the component name (com_jfbconnect). The file type shouldn't be a problem, and as you've noted, it should be UTF8. We do store our other language files as UTF8, but not the English version, so you may want to try copying the French file over to /languages/hu_HU/hu_HU.com_jfbconnect.ini and modify it from there.
Hopefully, you already figured it out, but if not, just let us know and we'll definitely help out however we can (and faster this time).
Thanks,
Alex