I was going to post this a while back, but never did for some reason.
The green ground wire for the track ball being used as a means of protecting the electronic circuitry inside is a somewhat silly notion. The boards are grounded and a good distance away from any metal parts that might be able to send any juice to sensitive parts. The amount of static energy required to jump those distances would be
huge, and not likely to be generated by spinning a plastic ball against a steel roller.
The ground wire is indeed there to prevent static energy build-up in the ball, but not to keep you from getting a shock either (although this is probably a side benefit).
A trackballs biggest nemesis is dirt and debris. It causes them not to operate smoothly, causes premature wear on the rollers and too much of it floating around inside the case can even mess with the optics over time. Nothing attracts dust and debris like a piece of plastic with a static charge. Debris sticks to it like it's a crud magnet.
So the real answer is: No, you do not
need to hook up the wire, but it is a good idea to do so if you want the unit to stay cleaner, perform better and prolong the life of your rollers.
Can we finally put this one to bed now?

RandyT