I checked the https demo using libcurl, and it failed unexpectedly with error code 35 (Unknown SSL connect error). I was expecting curl error 60 (untrusted certificate).
The test site requires SNI and only accepts TLS 1.1 and 1.2 with ECDHE/DHE+AES-GCM/AES-CBC ciphers[0]. I'm guessing whatever SSL library your libcurl is linked against only supports TLS 1.0, doesn't support SNI, or doesn't support any of those cipher suites (OpenSSL 0.9.x won't work with this site, for example).
[0]: https://www.ssllabs.com/ssltest/analyze.html?d=helloworld.le...