I have gotten the eyes to rotate with constraints but I had to change the way the contraints are set up. I set up the x, y and z axes to have constraints on local space. When I set up the constraints in world space, the eyes are facing in the wrong direction. I must have done something strange early on in making the eyes because when I first parented the eye bones to the head bone, each one jumped out of the head to behind it on the y axis. I merely set the origin to geometry and that seemed to have solved my proplem until I got to constraining the eyes in this lecture.
Here are pictures of the eyes first in local space and second in world space:
