I'm not exactly sure what you mean.. JFBConnect works with a Facebook application. That application has to be created by a Facebook user. JFBConnect can connect to your Facebook Page in multiple ways by showing your Page stream on your site or showing your site inside a Facebook Page Tab. However, when user's login to your site, they won't become members of (or automatically like) your FB Page. Facebook has no way to do that.
Hope that explains, but let us know if you have any other questions,
Alex