I have a toy poodle, chicks and grown chickens. Yes, the toy poodle is an excellent watch dog, but that doesn't mean the dog is going to be aggressive toward chicks. It is all about proper introduction. On the other hand, a cairn terrier is an obstinate dog that is a very independent thinker and less likely to control his instincts towards a moving little ball of fuzz.
I would invest in a predator proof coop instead of relying on a terrier to " protect " my flock.
Now, having said that, here are some tips that might help you lower the dog's urge to run after and catch the chicks:
If you play with them using squeaky toys, stop doing that right away.
Don't seek your dogs to run after moving small animals, birds, small children( during play), etc. Moving "targets" should be treated as things to be ignored. Walk your dogs the other way instead, enticing them with calm words to pay attention to you. This can be done by offering a treat, making an interesting noise, offering a toy they really like. You know your dogs.
The poodle is a highly intelligent dog, highly connected with its people, so it shouldn't be hard to show him proper behavior.
The cairn terrier is a fun loving dog, easily trainable IF its people know what they are doing. Distractions w fun work well with this dog.
Remember to never act anxiously around smaller animals and always have your dogs on a leash while training. It is easier to avoid undesired behavior then by having to control an overexcited dog.
Another thing to remember is to have a protected fence, gate, something to be a visible barrier between your dogs and the chicks. Holding a dog on a leash to close to the chicks so as it gets excited and the chicks overly stressed out is not going to bring a good outcome.
Also, dogs can feed from each other's excitement. It is always better to train your dogs separately and watch their behavior and improvement before having them together near the chicks. Dogs are pack animals and it doesn't take long for them to feed off each other's impulse.
There is a lot more to training. These are just tips. If you'd like more info, I'll be glad to help.