Thanks for all your suggestions!  I did some significant work on the eyes, and I think at least some improvement has been made.  I moved the slight redness to the lacrimal's side of each eye and darkened the outer edges of each iris.

This is what the eyes look like with a cornea trans map, a tear trans map, the highlight color on the Sclera set to gray and the highlight color on the Iris and Pupil set to black (all three highlights also have the main texture map applied).  The EyeSurface has no maps applied and has the color diffusion and highlight color both set to white.  There never were any reflection maps or painted-on reflections involved, and that hasn't changed.

Please let me know what you think.