Hi Slumsnake,
Here’s a simple way that looks to work with any/all animations.
First, make sure your eye mesh is a child of your Character Mesh. (Drag + Drop items inside of the ‘Components’ panel)
Then you can set the parent socket for your eyes to ‘head’, and then give them a location offset to put them where you want.
For a ‘glowing’ effect - in the eye material, multiply the texture/color by about ‘100’ and feed it into the ‘Emissive Color’ node.
The same attachment method should work for FX if you wanted to attach FX systems to specific points.
Hope that helps! Welcome to the Forums.