Well it's a puzzle for sure. I installed Perl and those modules with the same version you have, and it just worked with the updated code. The old code gave the same Net::SSLeay debug output as yours. There is some underlying difference between our setups obviously.
There's a chance that changing the SSL version to this would help:
SSL_version => 'TLSv1:!TLSv11:!TLSv12:!SSLv2:!SSLv3'
The last two components are supposedly redundant. But in any case, that long version string wasn't needed here (with any of 3 machines).
Update: Just as a debugging step, try adding random garbage to the string you pass into SSL_version. You should expect the following error, otherwise something is up in Denmark:
Error: 500 invalid SSL_version specified