Hi line_noise,
At first glance, I'd agree with good monk jeffa - sounds like your Squid proxy may not be explicitely config'd for https.   But that's just a guess.

"(code) LWP with auth proxy" is a recently-posted tidbit using LWP with proxy.   I'll tweak it for https and try at home this weekend from behind my Squid.   Will update this node with results and /msg you.   If I can find pertinant Squid config params will include them as well.
    cheers,
    Don
    striving toward Perl Adept
    (it's pronounced "why-bick")

Update:
Looks like igorash hit the nail on the head 8^)
A Google search for LWP https pointed me at Crypt::SSLeay, which is described as "OpenSSL glue that provides LWP https support.   Syntax examples at http://search.cpan.org/doc/CHAMAS/Crypt-SSLeay-0.25/SSLeay.pm