Since you listed Crypt::SSLeay, you want to turn off your proxy. I played with this for a while with good results. I used my credentials for PAUSE, but you can use any https site. Give it a try.

#!/usr/bin/perl

use strict;
use warnings;
use WWW::Mechanize;
use Crypt::SSLeay;

$ENV{NO_PROXY}='*';

my $mechanize = WWW::Mechanize->new( 
                noproxy => 0, 
                stack_depth => 5,
                autocheck => 1
);
$mechanize->cookie_jar(
    HTTP::Cookies->new( file => "mechcookies.txt", 
                        autosave => 1 ) );
$mechanize->proxy( [ 'http', 'https' ] );
$mechanize->get_basic_credentials( 'https://pause.perl.org', 
    'abc', '1234567' );
$mechanize->get('https://pause.perl.org');
print
$mechanize->status(), "\n",
$mechanize->content_type(), "\n",
$mechanize->base(), "\n",
$mechanize->is_html(), "\n";
$mechanize->content( format => 'text' );
$mechanize->save_content('text.html');
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