I am doing this in my game by creating a separate skeletal mesh with one eye in it, then I put two of those into the character blueprint. If you scale the component the eye rotation will happen in the new oval shape. Add Eggshell eyelids and things look even better
I was a rigger at pixar for 4 years and we did something similar for toy characters with simple eyes. Woody, Buzz and many other characters essentially all have the same eye rig it is just fitted and shaded differently. The rigging systems are completely different, but its the same basic principle, its all about the deformation/execution order. In one step the bones do their deformations as though they are unscaled and in another step unreal is scaling the component, because of the order of operation you maintain the oval shape.
This does mean you will have to work a little blueprint magic, but if you make your eye blink and eye dart animation procedural its pretty darn easy to setup.
Let me know if you have any questions
