I don’t use unity, but after a quick read-up i think the best fitting equivalent to gameObject would be AActor/Actor.
The simplest object that can have a location and rotation in the level is an actor (in C++ called AActor, in Blueprint simply Actor). Ingame it has no visible parts but can be moved around and other actors (like characters) can get its location and do something with it (run to it, rotate towards it, etc).
You can add components to an actor, such as StaticMeshComponent or SkeletalMeshComponent to add a visible mesh to this actor or other components that are not visible but do certain things (like collision hitboxes). These components can add additional functionality or attach visible objects to that actor.
If i understand you correctly, you probably want to make a Blueprint that inherits from the Character class, then change its Skeletal mesh to whatever your skinned mesh is. Then you need to add a Actor reference in your Blueprint variables, which is then set as the actor your characters eyeball meshes are supposed to rotate towards (rotation would probably be done each frame, aka in the Tick() event/function of the character).