That's interesting because moving the eyes has no direct relation to speech. It could be something else, like clenching fists or moving the head. It seems that performing other motor movements during speech can help overcome a block. Investigating the causes of this could be quite challenging, as we would need to map what's happening in the entire body and brain during these moments. But I would guess it has something to do with the idea that by performing another movement, the brain can "unclog" the speech muscles during some neurophysiological process. This is entirely my guess, without any evidence.