I just went out there again and now I'm wondering if she really isn't sick. I took her out of the nesting box and set her on the floor and she just kinda plopped down to sitting position. She is very very calm. Way more than normal. I took her outside to the water feeder and she didn't want any, so I dipped her beak in it and she drank a little. All this time still sitting down. The dog even came over and sniffed her and she just sat there. Normally she would run from him. Then I took her back in the coop, put her on the floor and got a handful of food and put it right infront of her and she wanted nothing to do with it.
I'm contemplating more that she might be sick...because you described her bum and vent as very messy with green/white poo...was it runny? How messy?
Broodies don't normally have a messy bum. Broody poo is very big, very stinky and little runnier, but still not the messy glop of diarrhea or soft poo caked into the bum feathers. Is this "normal" for her? She might have vent gleet...a yeast overgrowth. Giving her yogurt or probiotics in her water won't do her any harm and might help, even if she is just broody.
I personally would wait a few more days, maybe a week, to see if she is sick before giving eggs or chicks for her. Broodies CAN act like they are sick because they don't move around, don't want to eat or drink, comb will look pale...but they shouldn't have a really messy bum BEFORE they start sitting....they can and often do mess themselves when they go in the nest instead of getting off of it...but it isn't the same as the look of poo running down from the vent soaking the feathers or lots of poo sticking onto the feathers because the bird is going too often and the poo is too soft.
If all looks well, and she looks just broody, then by all means give her eggs or chicks.
I personally do better with hatching eggs in winter/cold than fostering chicks as feed store chicks have gotten used to a heat lamp. I find I have a couple of days transition time before the chicks are comfortable with the idea of running to the hen for warmth...some run right into her feathers, others are hesitant and can hang back...that's when they get too cold, and some I've had have succumbed to the cold before they adjusted. It was just too much of a shock going from hatchery to store to another place to adjusting to a hen.
Many use a heat lamp in with the hen while fostering chicks so that they can run to familiar light warmth while they grow used to the hen and learn to obey her clucks. I burned a coop down with heat lamp use, so I don't use heat lamps, hence I only foster in the summertime and hatch in the winter (and spring, and summer).
My experiences and thoughts.
Good Luck with her...hopefully she is just broody and you'll have great success with eggs or fosterlings.
Lady of McCamley