You have other issues, but this particular one is triggered here:

sub login()
{
    $mech->get($loginPage);
    $mech->set_visible($email, $pass);
    #
    # the original line, below, has double quotes nested
    # within single quotes
    #
    #$mech->click_button('value'=>'"Sign in using our secure server"')
+;
    # no error if you remove them
    $mech->click_button('value'=>'Sign in using our secure server');
    
}
[download]

For whatever reason, this "Can't call header.." error is what WWW:Mechanize does if you pass a value for a button that doesn't exist in the request to the click_button method. Seems like a bug to me.

Indeed..it is. click_button does not check for zero matching inputs. Until that's fixed, you can trap the error yourself:

eval {$mech->click_button('value'=>'does not exist')};
if ($@) {
    warn("Can't click button\n");
}
[download]