I created vertex groups to assign color to the eyes. Maybe it’s better I used Cycles, but that was the only purpose.
There is no reason to do it that way, just add materials to the list, in EDIT mode select the faces you want to be one of the materials then hit assign and repeat for the others.
Also, I want to be sure the eye doesn’t possibly “exit” the eye socket with rotating on the Z and X axes.
As long as the ‘eye’ objects origin is properly placed and the eye bone is in the correct position it won;t. The eye you have now will be fine, I did notice your eyes have a double mesh, like there was a solidify modifier on them at one time, unless there is a reason for that go into edit mode and delete the inner one on both eyes.