Brotherhood squires are children serving in the Brotherhood of Steel. Technically the lowest rank one can hold in the Brotherhood, squires are not allowed to participate in combat operations. Instead, they perform errands and menial labor, occasionally participating in field training exercises.
Squires roam the main deck of the Prydwen. Upon interaction, they will comment on life aboard the ship and in the Brotherhood. During the assault on the Prydwen in the course of Rockets' Red Glare, the player can convince the squire that the raid on Cambridge Police Station was a drill or conflict of intelligence.
During Leading by Example the squire will accompany you while you clear out the hostiles in an area.
One of the squires can be found drawing a rocket and a cat on the main deck of the Prydwen.
The conversation with the squire during Rockets' Red Glare seems to have been intended to be cut content, as it is not fully voiced and takes place during the massacre aboard the Prydwen, well after the player makes their hostile intentions known.