Directly manipulating how you use eye contact is about sending confidence/authority signals. In an interpersonal context you can use direct (but not challenging) eye contact to give the impression of competence, authority, and confidence. Combined with warm expressions and verbal assurances of patience, it can be useful for keeping the spotlight on a shy player while giving them the feeling that they can take the time to think and choose their course of action.
With a disruptive dominant player, it may be useful to get their attention as a band-aid measure, but beyond the immediate disruption other measures will be necessary (like taking them aside and discussing the problem, or not asking them back next session, etc.).
It's worth noting too that eye contact can be a great way to add a dimension to the personalities of NPCs. Direct eye contact, erect posture, and limited head movement can give an NPC an air of authority; shifting eyes, shifting posture, and refusing to look at the players except fleetingly can give the NPC an air of nervousness or otherwise lacking confidence.
(There is a chapter in Graham Walmsley's book, Play Unsafe that specifically covers using eye contact and posture to subtly and not-so-subtly communicate authority when playing characters.)