OK, I've got some more information here. ssltest.bat is just the output of 'pl2bat ssltest.pl'. More importantly, I am going through a proxy server -- that had me stumped for a while.

So what seems to be the issue now is that my proxy server isn't accepting my requests. Apparently there are two ways of doing this. The first is what LWP::UserAgent uses if I set $ua->proxy(...):

GET https://www.secure.com HTTP/1.1
[download]

and the second is what Crypt::SSLeay uses if I set $ENV{'HTTPS_PROXY'}:

CONNECT https://www.secure.com HTTP/1.1
[download]

In the first case, the proxy returns '400 Bad Request', and in the second it returns '403 Forbidden'. What that seems to mean is that the proxy doesn't know what to do with a GET request for a remote site, and while it does know what to do with a CONNECT request, it's configured not to allow them.

Does that seem correct?