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";
[download]

Does this work for you?

Edit: corrected, removing the trailing slash from proxy url