Here's a simple Toon Eye shader with some nifty eye related features. The pupil and iris centre is pushed back via parallax, and the specular reflection is a static matcap, so it always glints in the same place. This is not at all realistic, but gives a more controllable toony effect.
I've split the input textures up into separate Pupil, Iris, and Specular textures to make it easier to swap stuff out, but these can absolutely all be squeezed into one single RGBA texture if you want.
UPDATE: I've added in the 'No Texture' version of the shader - this one derives all the colours and shapes from mathing up the UVs and normals. Is this a better way to do things? No, probably not. But, it's interesting! Something to to have a look at hopefully.

The 'no textures' version.
Also, with a bit of tweaking (using actual normal maps) I think I could convert this into a nice *realistic* eye shader with all the bells and whistles. So I might make a version of that soon too.
Any questions, suggestions or comments, let me know here or on Discord. Thanks!