I have a blue heeler, and she is very motion-activated. Chickens moving slowly...okay. Chickens startled by curious dog...shriek-barking and lunging chaos!
Blue heelers are bred to herd. Some have very strong prey drive. Virtually all of them I've met are challenging (though rewarding) dogs.
While training, establishing strong leadership, giving the dog another job, etc. etc. are all helpful, none of them will rid a heeler of his or her natural tendencies. I think that if the dog running is incompatible with your chickens running, finding a new home for the dog might be the best choice. It's your choice based on your priorities, and whatever you choose is valid.
Our dog only gets to run free when the chickens are in the straw yard, and otherwise, gets to see them from the end of a leash. It's a compromise all participate in.
One other consideration: One dog might not think killing chickens is great fun, but sometimes they act like wild teenagers when they get into groups. Is there any possibility that only the heeler is catching them (and getting blamed), but the other dogs are participating? Doesn't change the outcome for either the chickens or you, but it might be worth considering.