Yes, you can use a different JRE version than the one from the OneClick page as long as it is a version greater than or equal to the supported version. You will also need to download and install the JCEUnlimited Strength files for that version. You will need to update the OneClick Client Configuration page to use a newer version of JRE.
Please keep in mind when using a using a JRE version other than the version which is provided to you could result in undesirable application behavior.
Regards,
Joe