To your question:
<blockquote>
How can I put the Facebook Connect blue button somewhere else on my site?
Simply use the {JFBCLogin} tag anywhere on your site and the Login button will appear for guest users. You can also include a logout button, when users are logged in, by specifying {JFBCLogin logout=true}. There are more options for the look of the button that can be seen in the Tag section of the JFBConnect Configuration Guide:
These tags can be placed in an article, a custom HTML module, or any template file. All settings in the JFBConnect component (new user type, redirections, etc) will be used automatically.
Source: http://www.sourcecoast.com/jfbconnect/docs/general/common-support-questions
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Hope this helps