Hahah wow I love the way those look. I think it depends on which character you're referring to. The main guy at the beginning only has a few eye shapes. As I kept watching, the duck character seems to be more complex with more geometry and real eye direction.
Animating UV placement might be a good way to get the eyes to actually move around a little bit. But for the ones where the eyes are simple shapes, doing set driven key on an image sequence seems like the way to go.