Almost every year I have flea beetles on my broccoli this is what I do to get rid of them....
6 cloves garlic
1 tablespoon crushed red pepper
I crush the garlic and the red pepper and place in a mason jar. I pour 1 quart of warm water over the pepper and garlic. I place this in the windowsill for about a week. I strain this and place it in a spray bottle. I spray on both sides of the leaves....I never do this on a sunny day it might burn the plants.