If you have no Linux experience, it would be best to start with Debian.
Seconded. The easiest way to run JRiver on Linux is with Debian.
I've never heard of amahi (and I haven't seen anyone mention it on the forums before), but it looks like one of those "web interface only" type custom linux setups that hides the guts of what's going on from the user. Those can be nice if you want to do exactly what they're made for, but if you want to customize them (say by running JRiver) they can be a real pain.
Assuming the Amahi does it's filesharing using Samba, you could probably access files on the Amahi server from another computer running JRiver fairly easily. The harder part would be running JRiver on the Amahi server itself. It looks like Amahi is fedora based, so you might want to check out the fedora install thread here for some ideas. JRiver needs a working x11 server, which amahi may or may not ship with (it's designed to run headless so it may well not), but you could potentially install and configure x11 yourself.
Or you could install debian and have a JRiver server up and running in a basic way in about an hour. There's nothing Amahi does that can't be done in Debian (filesharing, vpn, etc.), you'd just need to configure it separately. Depending on which features you need, that might take a little or a lot of time, in part based on your linux comfort level.
If you decide to try and get it running on Amahi, I'll be interested in how it goes (and if you succeed, you should write up your method so others can follow!). I'd offer to help, but I don't personally run fedora anything, so I could only provide generalized assistance at best. If you do go the Debian route, I can probably be more help, as I'm running JRiver on a homemade Debian server right now.