Hey Snow,
This shader is quite specific and there are settings exposed within the material instance which control the specularity for specific parts of the eyeball. You can modify these settings as well as create a new light actor and set it to a specific light channel. You can then set eyeball to that light channel so it adheres to the light settings and does not affect other objects in your scene.
Let me know if you have further questions or need additional assistance.
Cheers,