These are the Plymouth Brethren. There are two sects, the Open and the Exclusive Brethren. The Exclusive are a smaller sect and one of their tenets is not to sit at table with those who are not one of them, basing this on a text from St James's epistle. Google Plymouth Brethren, Open Brethren and Exclusive Brethren for the full history.
It just seems odd that they send the child to school but not allow him to eat with the other children. He eats in the school office with the office staff eating their lunch !!