Hay would not cause the green poop, hay should be a chinchilla's primary food (70% or more of their diet) and chin's poop should be dark brown/black in color not green. In fact a hay only diet can actually improve the quality of poops in chinchillas. If you are feeding her fruits or vegetables or other inappropriate foods or unhealthy treats that would cause it though. I would stop treats all together and see if that improves things.
Well the clean and cozy is not likely causing the issue you are seeing that doesn't mean eating it is safe at all. The issue with those paper litters is they expand when wet so can cause a gut blockage if eaten, meaning small or no poops not green poops. If she is eating it I would change it out with a safe option, like kiln dried pine or aspen shavings or fleece, but keep in mind fleece is also not safe if eaten.
Tap water is not safe for chinchillas, not only can it contain all kinds of chemicals, it can contain parasites like giardia. Giardia can cause gut issues, the first sign can be the poop turning green since the gut is trying to work to rid itself of the parasite, which will turn into diarrhea if left untreated. If the poops don't return to normal by stopping treats, or if they start having a smell to them (they should be odorless), become soft or wet you need to take the chin to the vet. Ideally bring some poops with you so they can test them.
I would also switch the chin to purified water, either buy bottled reverse osmosis water or get a filter that filters out micro organisms. In the past I had a water pitcher that had a built in filter, but the ones you put on the tap work too. Just make sure it's specifically says on the filter that it removes microbial cysts like giardia, not all do. If she already has giardia switching water now wont make it go away, you need to get anti parasitic meds from the vet, but it will stop her drinking in more. If she doesn't have giardia yet then switching now will prevent her from getting it.
