Run the one-liner:
perl -MWWW::Mechanize -e 'print $WWW::Mechanize::VERSION."\n"'
to check your WWW::Mech version. It runs fine using my 1.20. Perhaps your version is too old to include this method? Try using one of the submit() or click() methods instead, if this method really doesn't exist.
Comment on Re: WWW::Mechanize click_button method not found