We built cages with nest boxes built in. We slip a board or screen in to close off the nest box when not in use, open it up when needed. Has been great.
You might consider just building them permanently onto the side of the cage (you can make them more secure than if you hang them), and using something to barrier them off when not in use.
We also made double cages - a board slips between two of them. So when Mama is pregnant, we close off everything (she's in the middle). When she's ready to kindle, we open the nest box. When the kits are coming out of the nest, we open the second cage. After they wean, we put either the growout bucks or does into the second cage and close it off again.
Nice system, keeps us with more cage flexibility.