I got it to work. This seems to be a problem with the way Joomla is including mootools, specifically the leading backslash. There was a reference to this in an earlier post:
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Line 47 in '/components/com_jfbconnect/helpers/facebookhelper.php' originally said:
echo JHTML::_('behavior.mootools');
I replaced that line with:
echo '<script type="text/javascript" src="media/system/js/mootools.js"></script>';
leaving out the leading backslash.
I also commented out line 38 in '/modules/mod_jfbclogin/mod_jfbclogin.php' which is another call to include mootools.
I'm not sure if this is going to cause problems elsewhere, but it did solve the disappearing button issue.