I don't know if the Share button will display without the Like button, but it's worth a shot. My guess though is that it still may not work.
JFBConnect loads the Facebook Javascript library onto the page. Facebook likes to track people and the Do Not Track feature in Firefox will try to block that library to prevent the tracking.
I just tested a bit more with Firefox. When I visit the competitor's site you mention in 'normal' mode of Firefox, the Share button does display at the top as well as the Facebook Comment box. That should be the normal behavior you see as well where the JFBConnect Share and Like buttons also show.
If I open a "Private Browsing" session, then the Share button still does display but the Facebook Comment box is missing.
Are you seeing the same behavior? Mainly, is the Share button missing only when you're in Private Browsing? Can you tell us what happens on this site? Do you see the Share and Like buttons when you're using Firefox in normal mode (not Private Browsing)?
Thanks,
Alex