I got a signed SSL certificate from the Jabber Foundation / XMPP Federation. It works in most browsers except Opera, IE, and Safari on Windows for the time being. Apparently there are political issues surrounding getting the CA cert into the Windows trust store, but it should be supported in the near future. Here is an official but partial list of supported browsers; one not on the list is Safari for Windows.
For those who are affected by this, here is how to add the root CA to your trust store and make secure connections work: PDF.
Big thanks to the Jabber Foundation for the the help!