I agree with AD....
The Eeebox is a low power machine, and some form of Linux is likely your best option. There are no "stripped" versions of linux specifically for JRiver. If you are new to Linux I recommend starting with Linux Mint, it will be much easier to install than Debian, and you can be up and running fairly quickly following the linked instructions, just copy and paste from your browser to the terminal. If you already have Windows installed you can install Linux in a separate partition to dual boot between systems (just follow the install prompts), and you can even add a couple of other Linux distros for multi-boot options.
https://linuxmint.com/download.phpStart with Cinnamon, IMO one of the most user friendly and Windows-Like desktop environments, good for newbs. After you get comfortable with that try one of the lighter desktops, Mint MATE or Xfce, or even Lubuntu. You may notice a slight speed increase with the Eeebox with a lighter desktop. An SSD would certainly enhance the experience if you already have one.
https://lubuntu.net/Don't be afraid to experiment. Give it a shot and ask for help along the way.....