Thanks Volker, particularly for the debug browser link. After reading your first reply, I ran up a little test case that simply instantiated a QSslSocket and, lo and behold, somehow QT_NO_OPENSSL was defined by my configuration. I am a fool and need not have bothered you.
Looks like I need to rebuild my Qt, after checking my configure output more carefully. As I'm rebuilding anyway, I'm going to define '-openssl-linked' and fail fast if OpenSSL can't be resolved.
Thanks again for your time and advice.