yes, but I wouldnt let them drink a whole bottle of it or anything. also would not let them have a lot of cookies or cake that has food coloring, purely for the fact that they shouldnt be having too much treats. and no it will not change the color of the eggs or shells.
It's probably about as good for them as it is for us. Naturally made ones (vegetable based) are probably better than synthetic. But in tiny quantities it should be fine. What experiment requires you to color your scratch feed?