I have just tested the current version with Firefox 3.6.13 (Linux Ubuntu) and JCE 1.5.7.6 and did not have any problems. So I assume, that the reason of problem might be somewhere else. But to be sure. Change the default editor to TinyMCE and retest. If the same problems occur, we are sure that it's not the JCE. I then assume, it might be a JQuery library or so. Check if the problem also occurs in the joomla backend.
To use it in the frontend change the parameter "No conflict (Mootools >= 1.2)" in the backend menu Components->SWFContent Pro->Global Settings and retry it again.
If it still doesn't work please copy the Javascript error in Firefox->Extras->Error Console Menu, by clicking in the context menu (right mouse button) on the Javascript error and post the error you are receiving.