You can certainly define "zones" in JRiver and output to them simultaneously by linking them. How in sync they'll be is uncertain depending on the type of device, but if they're simply stereo pairs from the same soundcard (a 5.1 soundcard for example that allows you to redefine the outputs and access them separately) then it should work well.
If you really require the remote speakers to be perfectly in sync all the time then the easiest way may be not to get JRiver (or any other program) to play to both zones/outputs simultaneously but rather just play to one output and then literally split or distribute the audio. For example, a 5.1 soundcard (or onboard sound) that allows you to define one stereo pair as front L/R audio and then redefine another output also as front L/R audio, then Windows should take care of that and output to both sockets at the same time. When you plug a cable into a socket the Windows driver should pop up a dialog asking you which speakers you've plugged in - simply reply front L/R to both the local speakers and the remote speakers. You can then take a cable from there to the other room. Or if cabling is a problem, use something like the Marmitek Audio Anywhere transmitter/receiver system to transmit wirelessly. All this would work with any Windows program. Of course, the output is line level so in the other room you'd need an amp or a pair of powered speakers. If you need more than one remote room, you can also get audio distribution amplifiers which take one stereo pair as input and provide, say, 4 identical outputs.