Brent, all the science says the one can NOT get the flu from the shot or the nose spray. Not that the chance is low, or it is rare, just not gonna happen.
Misconceptions about Seasonal Flu and Flu Vaccines | Seasonal Influenza (Flu) | CDC
I know many people will never believe what the government says but that does not change the facts.
Some years it does a much better job of keeping the flu at bay and some years it does not work very well.
You are a man that nothing will change your mind about it. It is like arguing about the use of seat belts. That science is settled as well and there are many people that refuse to wear seat belts because their mind is made up. To quit your job over a shot is silly in my estimation but it is your job to quit. Good luck with whatever you decide to do.