The only difference between JRE and non-JRE is that in the first case jre folder is bundled in the package (you can find it in the installation folder). Which contains Java Runtime Environment, simply Java. You can install and keep updated your own installation of Java on the server, if you wish. Using JRE bundled is easier and it is not affecting other apps on the system that use Java. This is your choice which one to use. Other than that both versions are completely identical. Plugins are not affected by that.
Yes, it would be a big jump, so maybe read on this Explanation: upgrading Openfire
2.7.1 should still work.