The following code works for me:

#!/usr/bin/perl
use strict;
use WWW::Mechanize;

print "Versions\n";
for (qw(WWW::Mechanize LWP::UserAgent)) {
    print "$_\t" . $_->VERSION, "\n";
};

my $url = 'https://www.edited.com/';
my $user = 'user';
my $pass = 's3cr1t';

my $m = WWW::Mechanize->new();

$m->credentials($user => $pass);

$m->get($url);

print $m->follow_link(text => 'Enter');

print $m->title,"\n";
print $m->uri,"\n";

$m->click_button(name => 'B1');

print $m->uri,"\n";
print $m->status,"\n";
[download]

It outputs for me:

Versions
WWW::Mechanize  1.52
LWP::UserAgent  5.814
HTTP::Response=HASH(0x11fea1c)Automated Splice Site Analyses
https://www.edited.com/cgi-bin/protected/disclaimer.cgi
https://www.edited.com/cgi-bin/protected/menu.cgi?menu_gene.html
200
[download]

So I assume it has something to do with the versions of the module(s) you have installed.