If you define $ENV{HTTPS_PROXY} = 'http://proxy:8080'; you must NOT declare the proxy to LWP. So the correct code is:

#!/path/to/perl
    use strict;
    use LWP::UserAgent;
    use HTTP::Request::Common;
    use Crypt::SSLeay;

    $ENV{HTTPS_PROXY} = 'http://proxy:8080';

    $ENV{HTTPS_DEBUG} = 1;


    my $myurl = "https://www.redhat.com";

    my $ua = new LWP::UserAgent;
    $ua->cookie_jar( {} ); 
    $ua->protocols_allowed( [ 'http','https'] );
    # don't use https proxy with LWP!
    $ua->proxy(['http'], 'http://proxy:8080');  

    my $page = $ua->get($myurl);

    die "Error $myurl\n ", $page->status_line, "\n Aborting" 
    unless $page->is_success; 
    print "Success", $page->content_type, " document!\n";
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Does this work for you?

Edit: corrected, removing the trailing slash from proxy url

I'm still always getting "400 Bad Request" errors when I use the https proxy. If I try the code exactly as written, I get 500 errors so it's not using the https proxy at all via the environment variable.

There was a slight mistake in my code example:

$ENV{HTTPS_PROXY} = 'http://proxy:8080';
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Note the absence of trailing slash. It works for me, I've just got it running through my local squid proxy:

[~/temp]$ ./test.pl
SSL_connect:before/connect initialization
SSL_connect:SSLv2/v3 write client hello A
SSL_connect:SSLv3 read server hello A
SSL_connect:SSLv3 read server certificate A
SSL_connect:SSLv3 read server done A
SSL_connect:SSLv3 write client key exchange A
SSL_connect:SSLv3 write change cipher spec A
SSL_connect:SSLv3 write finished A
SSL_connect:SSLv3 flush data
SSL_connect:SSLv3 read finished A
Successtext/htmlcharset=ISO-8859-1 document!
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Please provide more information on your configuration. It works fine for me on Linux, running Perl 5.10.1 and LWP 5.835.

use Net::SSL (); # From Crypt-SSLeay
use LWP::UserAgent;
$Net::HTTPS::SSL_SOCKET_CLASS = "Net::SSL"; # Force use of Net::SSL
$ENV{HTTPS_PROXY} = 'http://98.142.214.160:443';
$ENV{PERL_LWP_SSL_VERIFY_HOSTNAME} = 0;

my $ua = LWP::UserAgent->new();
my $req = HTTP::Request->new('GET','https://twitter.com');
my $res = $ua->request($req);
print $res->status_line;
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