The driver is responsible..... I would check before I paid a dime. His insurance should cover it, at worst, it is a road hazard..... out here, people hit wild turkeys and they can cause damage if they come through the windshield. No one is responsible for that. IF they could prove that there was a chronic problem of chickens on the road constantly, then you might be required to secure your chickens, or any other animal, but at least in SD, it is accepted that animals sometimes get out, and the driver is expected to maneuver his/her vehicle to miss them.
But different states have different laws
But different states have different laws