Do I understand you right? You have a jre plugin for Firefox installed?
If this is a complete JRE installation you should be able to use this JRE for Phex too. You might just need to find out the JRE installation directory and call inside the phex installation directory something like:
/path_to_jre/bin/java -jar lib/phex.jar

Edit
Try to call:
/opt/jre1.6/bin/java -jar lib/phex.jar

I assume /opt/jre1.6/bin/java is just linked to /usr/lib/jvm/jre-1.6.0-openjdk.x86_64/bin/java

Apparently the java in your env path is a 1.4.2 installation, even though you have a more up-to-date installation available.

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