Egg shell color comes from pigments excreted in the shell gland. Very little she eats will matter unless one that were supposed to lay a dark egg was deprived of the nutrition that helps her develop those pigments.
It's like a dog eating bovine testicles. It won't make the dog more virile.
Food coloring may change the color of the feces but what they eat has to go through a lot of internal systems before it becomes an egg shell which has nothing to do with the digestive tract.
They do have one exit opening, the cloaca, but the digestive tract isn't connected to the reproductive tract.