Pabdi,
Glad you fixed the Artio issue. Should you run into any other login type issues, just let us know.
As for the Like button, it looks like Facebook has changed something on their end.. and the count is actually broken. Usually when they make updates that break things, they get fixed in 24 hours or so, so all I can recommend right now is waiting it out. For reference, our JFBConnect demo page uses the box-count style too, and the count is currently showing either:
www.sourcecoast.com/jfbconnect/
Additionally, if you use Facebook's own Like button configuration tool, you'll see the box count isn't working on that page either:
developers.facebook.com/docs/reference/plugins/like/
So, hopefully that helps explain, even though the solution is simply to wait.
Good luck,
Alex