Does Negative reinforcement work on dogs
This is a question that does not have a definite answer, and is certainly a controversial topic among dog owners and trainers.
The short answer: Yes, it works on dogs, but you shouldn't use it on dogs today.
In the modern era, positive reinforcement for dog training is a far superior alternative to training dogs.
Positive reinforcement is also an ethical way to train dogs, which is a must if you want to bond with your dog, rather than just work them to death.
In the past, negative reinforcement was the de-facto method of training a dog - not because it was effective, but most likely because there was nearly nothing on the topic of training dogs in the first place. (source)
We could argue that not everyone was a believer of negative reinforcement, but it's only speculation and there aren't hard facts that point to that.
What is negative reinforcement?
To start off, let's begin with the textbook definition of what is negative reinforcement for dogs.
Negative reinforcement is a type of operant conditioning in dogs that occurs when an aversive stimulus is removed after the subject performs the desired behavior.
To put it simply - you do something that the dog doesn't like (or potentially using pain) until the dog stops doing what you don't want it to do.

History of using negative reinforcement for dog training
The history of dog training wasn't all too long ago - it began only in the early 20th century, and it was to train dogs for war.
Despite our shared history of dogs, estimated to run thousands of years old (source), we still didn't understand our canine companions as well as we should.
A rocky beginning with learning to train dogs
Negative reinforcement that was used then were basically known as Traditional Methods.
Negative reinforcement and punishment were the two most common ways to teach dogs new behaviors.
Choke chains, prong or pinch collars, and shock collars were the main tools utilized.
Thanks to Pavlov and his experiment operant conditioning in dogs, traditional dog training became heavily based on that ideology - training dogs via operant conditioning.
To teach “sit”, a traditional trainer may pull up on a choke collar to encourage the dog to sit, and when the dog complies, the trainer releases the pressure.
If the dog complies, the pain (in this case, the choking) goes away.
It can simply be boiled down to, “do it, or else.”
Unfortunately, this traditional mindset of classical conditioning dogs still thrives today in some parts of the world.
A brighter future for dog training
However, after World War two, a new way of dog training has developed, based on new observations and studies of the dog’s natural behavior, and what it means.
For example, thanks to these studies, we now know in common knowledge that a bow indicates play, tail wags generally mean the dogs are happy, and so on.
This kind of training is often referred to as Pack Theory - aka Dominance based training, or more commonly dog whispering, popularized in modern culture by Cesar Milan.
The theory states that the Alpha Status is the highest rank in a pack, dogs develop dominance hierarchies in their packs.
To have control over your dog, you must achieve the Alpha Status.
This became popular in the 1970s and 1980s, and unfortunately, is still popular and frequently misunderstood.
One of the biggest issues with this theory is that it's based on popular belief, and these ideas are based on research done on wolves.
But, new data has revealed that dogs and wolves are not nearly as similar as previously thought, making the entire theory seem less credible.
Thankfully in this modern era, we've finally pushed the boundaries of dog training and learned that positive reinforcement training in dogs actually work far better than the previous traditional operant conditioning in dogs.

Negative reinforcement vs punishment in dogs
Negative reinforcement is basically training a dog to not do something, which can leads to STRENGTHENING of a certain behavior.
For example (not considering the morality of this action):
Let's say a dog is constantly barking.
Modern negative reinforcement training is done with the use of low level electric shock collars which do not cause injury or pain to the dog in any way.
The stimulus used is an unpleasant electrical current (either by vibration or sound) that you can turn on or off as needed.
This stimulus can be turned off as soon as the dog stops barking, leading to the STRENGTHENING of that behavior that the dog will stop barking unnecessarily.
Punishment is the REMOVAL or GIVING of a stimulus that WEAKENS a certain behavior.
For example (not considering the morality of this action):
Let's say a dog is constantly barking, the same as above.
A punishment can be shouting at your dog to keep quiet whenever it's barking. The dog, being scared of you, stops barking when you shout at it.
This stimulus is in your raised voice - every time your dog barks unnecessarily, you give your dog a stimulus, that causes your dog to WEAKEN the behavior of barking unnecessarily.
These are the differences, and both are not great ways to train your dog, or treat your dog either.
Why does negative reinforcement work on dogs?
After the explanation above, why then, are people still using negative reinforcement on dogs?
The answer is simple: Negative reinforcement for dog training DOES work, and can be used to correct bad behavior.
Does it mean that it should be used?
Perhaps, and only in certain situations, and very sparingly.
In the case of an already aggressive dog, using negative reinforcement techniques such as blocking, redirecting, or leash corrections, can eventually be used to bring the dog to a mindset where positive reinforcement will work.
Even this method is slowly being superseded by positive reinforcement training methods by trial and error.
The dangers of using negative reinforcement with dogs
The biggest reason is simple - Negative reinforcement training can destroy your dog's motivation to train (and yours, too)
Training should be fun for both dog and owner - when your dog is having fun, they’re willing to work harder and more often.
Waiting for your dog to make a mistake so you can correct them doesn’t sound like much fun at all, for you or your dog.
In fact, you'll become so focused on your dog's mistakes that you might overlook all the good progress your dog made.
It's also neither fun nor rewarding for your dog, and will basically not have any motivation to want to train, either.

Conclusion
Ultimately, operant conditioning is providing your dog with consequences for it's behavior - good or bad, from your dogs perspective.
Remember that the difference between negative and positive reinforcement:
Positive reinforcement provides something your dog likes to increase the chance of a behavior recurring in the future.
Negative reinforcement employs removing something your dog finds unpleasant to possibly prevent a behavior recurring again in the future.
Both are utilized to make a behavior more likely to occur again in the future, but negative reinforcement techniques for dogs are already being phased out in favor of positive, loved based training for dogs.
If you're considering using the classical conditioning for dogs, I strongly advise you against it - pick the positive reinforcement method and you'll see how much better it works for you and your dog.