I don't know why a drilled square hole would be useful.
Often required when a post is supporting a cantilever load. (e.g., a free swinging gate). A round post will eventually sag as the soil is displaced away by the force acting on it. A square hole (actually, the flat side) has a lot more surface area reacting the lateral force. "Square" is frequently specified by soil engineers but often ignored by contractors simply because they only have a round drill or auger. Nobody wants to spend for hand digging.