I found a workaround for this issue - not really a solution, but I'm in IT so it's close enough.
I was using Java 7u51.
I went into control panel -> Java -> Advanced
Under Advanced Security settings I selected:
Use SSL 2.0 ClientHello format
Use SSL 3.0
Use TLS 1.0
(those happen to be the types of connections the server accepted)
I also went into the security tab and added the site to the Exception Site List
Lastly, I went into the Java tab and clicked View. I then deselected Java 1.7.0_51 from my installed JREs.
I'm not entirely clear on how/why all of this worked when uninstalling & disabling other versions of java didn't. But I'm glad to have the site up and working again.
Hope this helps someone else in future.