Hello, I have it "fixed", but it's odd and it relates to the update only as far as I can tell. I am documenting, and I filed a bug with Apple via other means.
In the meantime, do this:
Download the full Mt Lion 10.8.3 installer.
Install the new full build (you won't lose data, just system updates).
After install, login and download:
Java for OS X 2013-002
and
http://www.java.com/en/download/mac_download.jsp?locale=en
Install Java from Apple, then Oracle.
Goto this page, and run the test:
http://javatester.org
Now, you'll likely NOT get the Java version info so click the "missing plugin" and then find the dialog that pops up allowing Java to run. (I didn't see this until I used Mission Control to show all windows) Allow and after the Java version loads correctly, you can then load your SSL page.
I would uninstall the Network Connect app and install again using the SSL page. It will give you the version I have above.
I am doing this again on another Mac to get the steps down and on paper. If you have questions, post back.
*If you keep getting the prompt to install the plugin or more info on the Apple page, just reload the page until you get the "allow this xxx to run" box and allow.
I've got this working now on 3/3 Macs. I know it's convoluted at the moment, so I'll try to get a better process/fix documented. Bottom line, you won't get SSL to load until the Java-Version info appears correctly in "pink".