"no reaction" doesn't mean they don't feel pain! Prey animals often do not behave any differently when subjected to painful stimuli, but that doesn't mean they do not feel it. They probably evolved to not show it because showing it would attract predators. Even if squawking and moving didn't attract predators, there would be no advantage for a chicken to show signs of suffering, because flock mates do not help sick and injured birds. They instead often kill them. People evolved to show it when they are suffering, because other people help them.
If chickens don't feel pain, why do they pull away and squawk when you accidentally step on their feet? Why do they make that little squawk and run away when pecked on by another chicken? If your vet cousin really said chickens do not "register" pain, that's an idiotic statement and I hope I never end up in your cousin's office with any of MY animals.
Sorry for the rant, but it's thinking like yours that people use to justify all sorts of animal abuse as seen in factory farm conditions.
Perhaps a more accurate statement would be that chickens don't always SHOW signs of distress when subjected to overwhelming painful stimuli. That appears to be true. But it certainly does NOT mean they FEEL no pain.
End of rant.