The biggest problem I would see with the arrangement is Chinchillas are nocturnal, chickens are not. So while the chicken is trying to sleep the chinchilla will be running, bouncing, etc. Also chinchillas will DIE in any temp over 80 or if the tempurature+humidity is over 150. I would reccmend chinchillas stay in the house, the basement is good if possible. They really the love to be between 50-70 degrees.
With wire bottom cages with chins be VERY careful. I had two chinchillas lose a leg because they broke their foot on a wire mesh shelf. I thought the first was a fluke. But after two, I don't have wire bottoms or shelving anymore. I paid $350 each time for an amputation. If they were not my girls pet..... Once they get caught on something they wig out and break their own bones.
With wire bottom cages with chins be VERY careful. I had two chinchillas lose a leg because they broke their foot on a wire mesh shelf. I thought the first was a fluke. But after two, I don't have wire bottoms or shelving anymore. I paid $350 each time for an amputation. If they were not my girls pet..... Once they get caught on something they wig out and break their own bones.