I worked with a company that gIve out thousands of flu shots. I hear it all. Inactive flu viruses(injections) can't give the flu. The mist can. It's live flu virus.
You cant get the flu from the shot. You can get the same kind of site reaction that children do when they get their shots. Make you fussy for a day or two? That's not the flu! Your kids aren't getting measles from immunization, but they may be sore and cranky for a day or 2.
Flu shots can't protect you from previous exposure. If you get the real flu within 2 weeks of shot, you were going to get it anyway. Last year's shot was not very effective. This year's seems to be better. The flu shot is preventive, not a guarantee.
I got my shot this year, plus tdap. I always have a day of yuk site reaction from flu shot. The tdap was awful. Arm hurt for almost a week. My entire family was coming down with a cold, but I got the mildest version, so the flu shot didn't disadvantage me. They are all young and healthy and I'm not. Go figure. I think that was just luck.
Your experience may vary, but you won't get the flu from the flu shot. You could get an allergic reaction. If you have ever had a serious reaction to a shot then you likely shouldn't get another unless you know the issue and eliminated it. Two yuk days are not the flu. Google what the flu actually is. It's serious and can be deadly.
My suggestion is not to give the mist to your kids. They can get ithe flu, albeit infrequently, from it. If you go to a good clinic that uses two needles, one to draw the vaccine into the syringe and another to inject, the needle is not dull and likely they won't even feel it. Don't dramatize immunization process. The least drama they get the less the will give. In clinics, usually the parents are the problem.