Facebook redirects me to: https://register.facebook.com/editaccount.php which I assume is the final destination. Assuming that WWW::Mechanize is set up to follow https redirects you should end up there too.
Much of facebook like most modern websites is generated by javascript which I think that WWW::Mechanize is unable to cope with, this could be the source of your problem. I recommend trying to scrape while appending "?_fb_noscript=1" to the url's you visit. This turns off much of facebooks javascript system and should enable a smoother ride for your little bot.
(Do remember this is only an idea and doubtless whatever you do is against facebooks usage policy and this makes you a "baddy." I don't condone such badness and beleve that baddies should be forced to go to bed early etc... ad nauseum)