Note that the error is origination from within WWW::Mechanize. It's not $mech that's undefined. WWW::Mechanize::click in turns calls the method click of another object.
The problem is that WWW::Mechanize doesn't find the button you want it to find but proceeds to try to click it anyway. According to the docs,
The first argument is the name of the button to be clicked
The document doesn't contain any buttons named Save/Reboot. You'd be better off using a method that can locate a button by value, such as click_button.
$mech->click_button( value => $button );