my two kids have egg allergy, and this is Year 3 of getting flu shots for them. The first two years we had the shot broken down into two parts, about 30 minutes apart. This year it is the full shot all at once. They've never had even a tiny reaction, surprisingly. We stay for 30 minutes after the shot. In fact, I'm taking DS today for his. DD had hers last week.
Yes, make sure you have your epipens with you; our clinic has epinephrine sitting on the counter ready to go as well.