Do you want to leave the Java application running, even after you disconnect? If so, install
screen allows you to create persistent sessions allowing you to keep programs/scripts running in their own session even after you log out of SSH.
If you're on Ubuntu or any Debian-based distro, installing
screen is easy:
# apt-get install screen
To create a session:
$ screen -R sessionName
Start your Java application in the session and then when you are ready to detach and let it run in the background, type Ctrl+A and then D. You can then safely disconnect from the SSH session.
To re-attach to the screen session, execute:
$ screen -r sessionName
The Screen man pages describe all of the other arguments and flags if you want to learn more.
If you simply want to save the output of the application for later reading, redirect
stdout to a file:
$ java -jar my.jar > output.txt