Only part of the answer:
Eevee materials that are transparent need some additional material settings turned on; for example, Blend Mode and Shadow Mode need to be set to Alpha Clip/Hashed/Blend (depends on the usage) in order to display transparency, and (I think it was translucency) needs Screen Space Refractions turned on both in the materials tab and the Render Context tab.
I don't quite know why the eye setup is so (unnecessary?) complex; but I noticed that there are a few material nodes that do nothing, like the eyeball texture is completely muted, two Translucent shaders (almost same value) are mixed with each other, and I also don't quite know why the eyeball texture is controlled by the cubic box, but it somehow contributes to the eyes showing up black in Eevee (though setting up my above suggestions partly helps). Someone else might have more knowledge about that, just wanted to note the basic transparency info.
EDIT: Okay, workaround: I copied the material of the cube and pasted it into the eyeball material (the one that only had a texture node controlling a Mix Shader with a Transparent Shader + nothing) and plugged the output of the last copied shader (Mix Shader) into the empty input of the Mix Shader that has the Transparent Shader connected to (the lower input). Changing the Blend Mode plus activating Screen Space Reflections then did the job for me. You can delete the cube then if it's not needed for anything else. Also, I would recommend turning down the mix value for the eyeball texture in order to have it visible.