After four failed attempts, the fifth was a success! With a double nut on one side to keep the rod from spinning, we were easily able to either pull or drive the bushings out with an impact gun. We used this method from then on, and it worked great!
Once the bushings were out, we cleaned up the inside of the holes with some sandpaper and Emory cloth so the new Hyper-Flex bushings could slide right in after using a light coat of the supplied Energy grease (As an aside, this stuff should be called glue, it is super sticky!).
Note, there are six total bushings per control arm per side — by far the majority of the bushings we needed to change, but not the most difficult (see rear differential below). Also note, there are four different bushing lengths and two different sleeve lengths, so make sure you are using the right ones in the proper locations.