Kids his age should be vaccinated and are classified as "high risk" for developing potential flu-related complications.
Managing an allergic reaction is far easier than managing the potential severe consequences that could arise with the flu. The risk for allergic reaction is small because the amount of egg protein is incredibly small.
The benefits of the flu shot far outweigh the risks even in patients with egg allergy. It was in the news not too long ago where some research came out about the flu shot likely being safe for egg allergic populations. The study followed a bunch of people with previous severe allergic reactions to consuming eggs and the rate of reaction was so small it was pretty shocking. Also, none of the reaction from the shot were severe and were basically only hives and such.
Hospitals require employees to be vaccinated and undergo annual TB skin testing. They do it because hospital patients often times have compromised immune systems and the flu can flat out kill them.
Kids of this age are at a high risk and the risk increases the younger you get. If this kid has contact with infants or other children under the age of two through an after school program he should be vaccinated. Even if they share classrooms and he doesn't contact these children directly, if he were to get the flu he's going to be spreading it all throughout the common area.