You should always run the most recent version of the Java Virtual Machine (JVM), unless otherwise stated on the Elasticsearch website. Elasticsearch, and in particular Lucene, is a demanding piece of software. The unit and integration tests from Lucene often expose bugs in the JVM itself. These bugs range from mild annoyances to serious segfaults, so it is best to use the latest version of the JVM where possible.
Java 8 is preferred over Java 7. Java 6 is no longer supported.
Either Oracle or OpenJDK are acceptable. They are comparable in performance and stability.
If your application is written in Java and you are using the transport client or node client, make sure the JVM running your application is identical to the server JVM. In few locations in Elasticsearch, Java’s native serialization is used (IP addresses, exceptions, and so forth). Unfortunately, Oracle has been known to change the serialization format between minor releases, leading to strange errors. This happens rarely, but it is best practice to keep the JVM versions identical between client and server.