I have Mountain Lion running via Parallels. I installed Phex, opened fine but no connection. Used the connection installer & Phex connected to its first peers in less than 5 seconds. I am in process of installing Mavericks to see what happens. The Java I had installed is the Apple Java, not Oracle's Java. I did that because LimeWire refused to open with Oracle Java.
I'll let you know what happens after installation is complete. Mind you, I'm not sure my version of Parallels supports Mavericks, but Mavericks is supposed to be very very similar overall to Mountain Lion.