There are insurance companies that will insure anything for a price.
Traditional insurance i.e home owners would not apply in this as it is land that is not in your name. And without a contract I would not think you could get renters insurance. Any experts please advise.
are you looking to insure the hens or the structure itself? if you're paying rent on an informal basis, then sounds like any liability would be on your friend (and their insurance). i'm not a licensed agent though - i just work with a lot of them.
I think it would be hard to insure against theft specifically because how would you prove it was (human) theft vs a predator? perhaps just upgrading the locks on the hen house would provide some peace of mind if you're really worried.
but you might see if you can find an insurance agency that specializes in farm/agriculture insurance and pick their brain about it.
I doubt either a homeowner's or renter's policy is going to cover the theft of chickens. Some farms that have very valuable animals take out policies on the animals themselves, not sure if such a thing would be available for chickens.