The latest version of openjdk and icedtea plugin work with Juniper VPN. Removing older and other versions of Java may also help.
See Could anyone provide a step by step for getting juniper netconnect and citrix?
This is one place where I found command line works better than the Ubuntu Software Center
First remove older versions of Java, both Oracle (Sun) and OpenJDK. For example, if you have OpenJDK 6 installed, type in a terminal:
sudo apt-get remove openjdk-6-jdk icedtea-6-plugin
Remove the Sun Java as well. If you have installed just the JRE, rather than JDK change the commands accordingly. Then install the OpenJDK-7 and corresponding plugin by:
sudo apt-get install openjdk-7-jdk icedtea-7-plugin
Open Firefox and test the version of Java is being used by going to this site http://www.java.com/en/download/testjava.jsp
If you see version 7 is running, you are ready to try Juniper VPN. If not you may have to remove the other version of Java. Hope this helps.
Update: for 64bit Ubuntu
From 7.3 onwards, Juniper SA devices support 64-bit Ubuntu for Network Connect.
First, install the 64bit jre/jdk and icedtea
sudo apt-get install openjdk-7-jre icedtea-7-plugin
Second, install the 32bit jre
sudo apt-get install openjdk-7-jre:i386
Note, do not install the 32bit icedtea. Make sure the 64bit is the default java. As long as the 32bit jre exists, Juniper will find the files it needs and run.
Tested on Ubuntu 13.04 64bit.
See Juniper Network Knowledge-base for details.