Re the egg / chicken question - a few years ago, I was working at a travel clinic, and a local business which imported handicrafts wanted to provide immunisations for all staff. After discussion with the clinic, the business decided not to go ahead, as most staff were buddhist and / or vegetarian / vegan, so not willing to accept products grown in eggs.
If you're allergic, you'd get a reaction within minutes or hours, much less likely to be days - and reaction wouldn't be a sore arm. A sore arm is pretty common, although less so with good technique by both injector and injectee - difficult to say to relax, but it really does work. Having been on both ends of the needle, I long ago realised that injecting into a tense muscle isn't comfortable! So, place the back of your hand on your thigh, then let arm flop. Some people do get cold / flu like symptoms after a flu jab, but so do many people who haven't had one - it's very difficult to know for sure whether it's coincidence or cause & effect.