This question is rather confusing. It's possible to click regular form buttons which you seem to know already. It's also possible to follow specific links. It's not possible to click JavaScript controlled links. Refer to the pod, WWW::Mechanize, and look for these methods. The site you linked is down/unavailable at this time.

  $mech->click( $button [, $x, $y] )
  $mech->click_button( ... )
[download]

Maybe what you need to do is manage your cookies better...? Check the pod for cookie info too.