- Consider a [track to] constraint which rotates the bones
You can track to the camera or some object like two rectangles next to each other lets call them a pair of eye trackables, similar in placement to a pair of eye glasses. Depicted above in the animation is an example. Left eye tracks left point, right eye tracks right point. The points are eye distance apart. At a far distance the eye trackables cause the eyes to rotate and be nearly parallel. At a close distance the this can prevent crossed eye problem. Some blender people might say the having a separate eye trackables is more flexible than tracking to the camera. You can move the trackables independent of the camera.
In the animation below the green arrow rotates to the red ball via the [track to] constraint. The other activity in the animation was used for another purpose prior to this question.Consider searching Blender Facial Rig Eyes
- Consider searching Blender Facial Rig Eyes