I am having problems using this with eevee. It would make the eye materials all blank white. How do I fix this?

You sometimes have to wait some seconds for EEVVEE to calculate the shading. Then the all white surface should turn to the material colors:

There is a coating over the eyes (Glass Material) that is disabled in the Viewport, but not in the Render, hence the difference. You’ll need to open an Eye in the Outliner and select the (greyed out) Coating and in the Material Tab enable Screen Space Refraction and set the Blend mode to Alpha Clip or so:

Also enable Reflection and Refraction in the Render Tab:

BUT, this wasn’t made for 2.8+ and the dependency graph has changed completely and this file gives loads of errors and/or warnings so the rig might not work correctly:

Thanks, @spikeyxxx, for the in-depth explanation.  The “Blend Mode” setting to one of the “Alpha” modes is only necessary for a “Transparent BSDF” which doesn’t need “Screen Space Refractions/Reflections” being activated:

Glass BSDF with “Screen Space Refractions/Reflections” and “Opaque” as “Blend Mode” :

The sphere in the right frame is only black because of the shadow being seen through it. This isn’t of course physically accurate but EEVEE specific and can be changed using a “Light Probe” object of type “Reflection Cubemap” in the scene for which you have to bake a “Cubemap”:

Tweaking the “Clipping Start” value so that it’s outside of the “Glass Sphere”:

With the “Reflection Cubemap” baked:

For “Reflection Cubemaps” see this video here (still works with Blender 2.91 although recorded with Blender 2.80 Beta).

 Important note from the Blender 2.91 manual here:

Screen Space Reflections are much more precise than reflection cubemaps. If enabled, they have priority and cubemaps are used as a fall back if a ray misses.

PS: @spikeyxxx  is there a setting to colorize the “WARN” error message or did you do it manually in the screenshot?

Thanks, @spikeyxxx  I’m slowly discovering EEVEE. It’s still in many parts a mystery to me with all the tweaking 😉. I think, I need a lot more pratice for getting really cool EEVEE renderings.