I'm severely allergic to eggs and have always been told NOT to get the flu shot. An egg-free version has supposedly been in the works for some time now, but I don't know that it exists yet.
The flu-shot isn't the only medicine that uses egg albumin as a base, some newer sedatives do as well. When I had a minor surgical procedure that I needed to be sedated for, there were seconds away from injecting an egg-based anesthetic into my IV. Thankfully my mother was there and stopped them before. People with egg allergies need to be especially careful with these things!