If your dogs had broken into the coop/run and killed the chickens, then yes absolutely you should pay for the lost birds. However, if the birds were not contained, the owner had to know that the flock was at risk of being killed by predators. It just so happened that the predators were your dogs.
I agree with this. If the birds were in your yard when the dogs got out, then chased them into their own yard and killed them ? Well the chickens were not contained in their own yard now were they ? $150 is ridiculous. This same thing happened to me and my neighbor asked 'what do i owe you' and i charged her $6 each, cause that's what i paid for the darn things ! If it had been a coyote who would she have tried to put the shaft to then ?