You're the first person I've heard have almost the same reaction I've had! I'm not allergic to eggs, either.
I had my first flu shot when I was pregnant 8ish years ago. I had a mild reaction - a little redness all over my upper arm. Next year, I had that plus it was hot to the touch and covered more of my arm. I just figured I was having a good immune reaction and it was worth not getting the flu. Year after that, I had all of the above plus the rash went across my chest and my stomach was a mess. I also got a flu that same year, and wound up with walking pneumonia from it. I didn't want to ever go through that again, so I went for the shot one more time and had all of the same reactions plus an asthma attack within 15 minutes that required medical treatment. We have no idea what the problem is, but I've been banned from any more flu shots, and we're not at all sure if other vaccines will be a problem, too. So I really work hard to avoid rusty nails, lol!