Just for reference, because I've answered the question already on Stackoverflow:

• with LWP::UserAgent 6.06 and LWP::Protocol::https 6.06 proxy support for https should work out of the box as documented, e.g. by setting proxy with proxy method or reading it from environment with env_proxy method
• for version 6.05 it might work out of the box if you use Debian or derivates, because they've patched it already in 12/2013
• for versions older than 6.06 Net::SSLGlue::LWP provides the necessary monkey patching