Not my first choice either TBH. And agreed, it's a method very easy for most of us to screw up. However the OP has described a behavior that will eventually get the dog killed by an irate chicken farmer if left uncorrected.
I also disagree that the dog is doing it as a form of play. Most chicken killing dogs are responding to prey drive. Positive reinforcement ain't gonna break that. If as you infer, it might be a "strong willed" dog and, as I stated, it's reacting to a strong prey drive, then it's unlikely any method will break it. That leaves three basic choices:
1) resort to drastic training measures as a last resort
2) take very strong measures to keep the dog confined to your own land, or
3) get rid of it.
My personal choice would be #2
Positive reinforcement is for reinforcing behavior you want them to repeat. Negative reinforcement is to break them of behavior you don't want them to repeat (extreme negative reinforcement is for a last resort to an extreme situation) Same as raising children; yes I spanked my daughter when she warranted it and still spank the grandkids when they do (rare occasions in both instances)
The problem with negative reinforcement is that the dog has absolutely no idea why it's being punished.
It's like when people punish their dog or rub the dogs noses in it when a dog poops indoors, after the dog has pooped it's completely moved on, so punishing it after the fact is pointless.
Likewise with Spandit's problem when do you activate the shock collar, when it looks at a chicken, after it attacks it?
You also need to look at a dogs natural behaviour, there used to be a dog on our street that used to get along great with cats, one day a cat from another area attacked the dog (pretty strong negative reinforcement) the dog went after cats from that day forward, the more cats fought back the worse it's behaviour got.
With positive reinforcement you are giving the dog something more interesting to do BEFORE it gets riled up, so you are taking that attack mode away.
I'm not convinced positive reinforcement would work either but don't want to resort to using shock tactics! We try to keep her in and when we get our own chickens, we'll keep her out, as we would for foxes. She's killed pheasant before and goes completely deaf when she goes on the offensive. She just likes killing things. Unfortunate, really but instincts are powerful things.
I reckon the only way she'd kill the owl is by cutting off the food supply by eating all the voles - the state of my garden/fields testifies to her success
Dogs have been breed for centuries to have a attitude that pleases us, i don't believe that if a dog is well fed it has a "natural" instinct to kill, even if it has it can be trained out of it.
As an example look to those poor dogs that are breed to fight, they're trained from puppies to attack and kill other dogs, yet with patience, socialisation and training this can be trained out of them.
The trick with positive reinforcement is in the timing, if you've found something that motivates her then you need to distract her with that before she locks on and goes deaf.
If that's not possible then try a different motivation, something really smelly like sausages can often do the trick as many dogs are lead by their noses.
Once you've found that motivation then use it to distract her as you walk past the chickens, the object it to eventually get closer and closer to the chickens without her noticing them.
With persistence eventually you'll be able to have her within a few meters of a chicken without going into attack mode, you'll then reward this behaviour, so she'll link chickens with good things.
It sounds easy but it's going to take a LOT of time, effort and patience, in my experience dogs just want to please us, the difficult part is getting them to understand what we want or in this case don't want them to do.
I don't agree with just fencing the dog off for the simple reason is that the behaviour will not correct itself, and by being fenced off the dog is likely to get only more frustrated, which just makes the matter worse.
It's also a very very bad trait for a dog to have, eventually it won't just be chickens and birds the dog goes after, it'll move onto cats, then other dogs, then faster moving objects like bikes and runners.
So it might only be chickens now, but IMO this is extremely serious and needs to be corrected.