I finally got the knobs to generate MIDI CC output! It turns out you have to send on channel 1 or greater. Channel 0 doesn't work. Perhaps the Arduino MIDI library subtracts 1 from them?
Anyway, I've got it attached to the SoftEgg Synth, the prototype of the synthesizer engine that will end up in the NanoEgg. I don't have all of the possible CC values set up in the synth software, but I do have about 16 knobs defined, and it works great with all of those.