Rear stabilizer is usually made of a steel bar with 4 connecting points, two of which are bolted to the frame and the ends are bolted to your axle. It may vary from vehicle to vehicle.
Rear stabilizer bar is not necessary, depends on what kind of vehicle you drive and your driving style. I removed my rear stabilizer bar and threw it into garbage after I lifted my 2000 Jeep Cherokee. I have been driving it without stabilizer bar for over a years now and have never had a problem.
But DO NOT try to remove front stabilizer bar,
Rear stabilizer bar is usually not expensive and easy to install by yourself. In your case, I do not think that your rear stabilizer is broken. Most likely, one of the bolts that connect the bar to either frame or axle rusted/sheered off.