Bill,
Sorry this thread got dropped. The Logout button is generated by JFBConnect in the following file:
/components/com_jfbconnect/libraries/facebook.php at line 306 in the function getLogoutButton(). This is called from the mod_jfbclogin/tmpl/default.php file at line 130. You're free to modify either portion to see if you can get something going, and let us know if you have any questions about the changes to make.
Unfortunately, I'm not really sure what's happening in your template beyond this. In the facebook.php file above, you can easily change the type to 'button' and see if that helps. The code in that block is used by the JFBCLogin module and the {JFBCLogin logout=true} tag, so once you modify it, it will change it wherever those are used (and hopefully fix all their styles).
Beyond this, I don't have any great suggestions. You may need to contact RocketTheme to get a better understanding of what's happening when the pop-up is displaying, and why styles are working on it.
Really hope this helps, and best of luck to you!