Well I configured one machine as yours to test:
LWP::UserAgent : 6.06 IO::Socket::SSL : 1.955 Net::SSLeay : 1.66
Just retried with SSLeay @ 1.55, still works fine.
This is all testing with Perl 5.14.2 to stay as close to yours as possible. However, it may well be the underlying OpenSSL version. All the machines here are Fedora 20 boxen.. with:
Would be interesting to see if your two machines have separate versions. By the way, on the machine that works for you, maybe give it a go with and without the SSL_version parameter.