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";
Does this work for you?
Edit: corrected, removing the trailing slash from proxy url