First, unless you have a site/s or an application that absolutely requires JAVA and you have no choice but view/use them then you should remove JAVA. It is is one of the biggest targets of malware and exploits are continually being found and require patching/update.
JRE 6 is pretty old and whilst still supported, JRE 7 is the latest JAVA stream, if you have JRE7 then you shouldn't need JRE6.
You can try resetting the software updater database - Software Updater > Settings > Database reset (troubleshooting), expand it by clicking the triangle > click the Reset database now button.
