I came in here to joke, but then realize you're posting questions about your kids so I'll be serious for a post.
Here's the deal...Flu Vaccine isn't supposed to make you sick, but you can have side effects from it. If you had the flu, you'd have serious temps and wouldn't bounce back from it quickly. Some flu infections can take weeks to clear and can be pretty serious if your immune system is down. Flu is viral, so antibiotics don't help with it....but they can help boost your immune system by fighting bacteria that may be spreading its defenses thin. Doctors that prescribe antibiotics too readily kind of suck and are just showing up to work at the expense of their effectiveness...(that's another rant)
In general, it's good to get plenty of rest and eat healthy through the Winter and Spring to keep your body healthy and defenses up...wash hands readily because Flu is spready through sneezes and coughs by people who don't cover their mouths...or DO cover their mouths, but don't wash THEIR hands.
Read this from the CDC...they conducted blind studies and found that flu shots can cause more site injection soreness, but otherwise doesn't cause much difference...
Read more on common misconceptions about seasonal flu and flu vaccines effectiveness.
www.cdc.gov
You CAN get a low-grade fever as a reaction to the shot. I would say, kids are more likely to have a reaction just because the ratio of vaccine to their body mass is different than an adult...
A lot of people die from the flu each year....it's why we bother with the shots. It's not something to take lightly or ignore. My wife works in healthcare and she gets upset when moms on Twitter/Facebook post like they're experts in vaccines and claim a vaccination can put your kids on the spectrum.
Sorry your kids had a reaction...just get them through it and hopefully it won't bother them next round.