IF you're desperate and are willing to accept 48khz mp3 at 320kpbs, then you can do it with additional software.
Probably not the scenario you're looking for though!
You can use two products from Rogue Amoeba: Loopback and Nicecast.
Create a Loopback virtual audio device, and then choose it as your system out in system sound preferences. Then, use Nicecast to broadcast locally (local internet radio station) at 48khz and 320kpbs, choosing the same Loopback virtual device as your input (i.e. system audio will be your input).
When you start the broadcast, you'll see a local IP address of your .m3u "station".
Use Open URL... in JRiver MC to open that stream.
Your Audio Out settings in JRiver MC will of course need to be something other than your system out, or you'll have... a loopback of a loopback?
It's a bummer we can't just have system audio in on Mac JRiver MC. Would love to have that.