As you can see in the picture, eye of kilrogg has been given a patrol command. The problem is, it does nothing. If you tell it to patrol anywhere, it simply won't. Even if you group a unit that can patrol with the eye, and try to patrol somewhere, the eye will still not move. It's probably for the best anyways...with a speed of 42 it would patrol back to where it started in a matter of seconds.
-Thanks to Stryker-17
NOTE: This was actually fixed in the first Battle Net Edition Patch!