When you say “joint box”, do you mean “Head or Eye area Joint”? That is the joint that will act as the base joint for your eye skinCluster, when it does the automatic weighting. That doesn’t make the rig follow that joint. In your video you show the joints, but it is your controllers you need to make follow.
For the rig to follow you still need to parentConstrain or parent the eye controls. I am pretty sure that the eyes_rigger doesn’t do that for you. I parent and scale constrain eye_L_root and eye_R_root to my own controls in the head, in a POST script.