There are several ways to approach , you can either:
a) make the eyes a separate plane that’s close to the head and use dynamic materials and change the texture based on what the character is doing
b) make the eyes have their own bones, in the animation blueprint use a transform rotator or IK setup to rotate them to face a specific direction (can be calculated based on player pawn and the other actor, like “find look at rotation” node for example)
c) make the eyes a spritesheet, extract them and create flipbooks, then on the character add it as a new component, attach it to the head, then in the blueprint create a new function like update flipbook or something along those lines, where it would set it based on whats happening with the character (happy, hurt, angry, etc.)
d) again with the eyes having their own bones, but add new keyframes to existing animations using persona
c) animate them in blender and export as part of the animation
e) look into control rig and have them rotate only during sequencer cutscenes
f) plan F, as in totally f… is to buy an iphone and check out the facial expressions demo and make a custom head with all the bones required and record new animations that way
There are probably more ways, but those were on top of my head how I would experiment with it.