How was it attached?
A 7 year old obese lab is over middle aged. It sounds like that wouldn't be easy for him. Welded wire is tough.
And when he killed the neighbors chickens, that was 4 years ago, when your dog was 3. A three year old lab is just out of puppy stage.
I just can't help thinking there must have been something different about that day. A fox that ripped it open, or something. As Islandgirl said above, he would have been tired after the first few, and the would have been running around like crazy.
There are so many factors, did you check his teeth? Did you check him completely for blood? Dogs tend to 'death shake' things (I'm sure you've seen how they shake their toys) so were the chickens necks broken? How big were the puncture wounds? Labs have more rounded teeth because they are retrievers, he would have had to apply considerable force and create larger holes than say, a cat, fox, or another breed.
A muddy dog is not a chicken killer. Maybe someone else's dog broke your fence, and your dog only attacked the rooster that was out of the pen because it was easy and he was excited then went to investigate. Maybe there was a cat in there and the dog broke the gate and then got over stimulated.
(also, a dog that looks guilty is usually just reading your body language and voice. I can make my dog duck his head in 'shame' by saying his name with a certain disappointed lilt. After you found the chickens, you were definitely mad and probably calling for the dog to come, but since it wasn't in a happy voice he cowered in the front yard)
Honestly , it doesn't matter what actually happened. We'll never know. I'd just like to know the dog is safe and will be getting chicken training instead of being rehomed. I don't want to turn this into a post about what you should or shouldn't do with your pets (or a personal attack, please don't take it as that) as you just wanted to know if he did it or not, but you've had the dog much longer than the chickens. He doesn't have a whole lot of time left. His chances in a shelter are slim. Giving him to a relative or friend is better, but is it really fair to the dog?
Even if the dog did it, please don't blame him for doing dog things. Even if my dogs tore into my chicken coop and destroyed all my chickens, I would consider it a failure on my part, for not training better, for not making it strong enough, for leaving the door open, and not a failure on the dog's part.
Yes...to all of this.I hope he didn't do it.....but even if he did please think about this.....your dog loves you. You are the only family he has known. All he wants to do, lives to do, is to please you. Please don't doom him.......
You'd be amazed at how small of a hole a fox can actually fit through, how much patience they have to wait for just the right moment and enough intelligence to look for any weak areas that would allow them the simplest and quickest way to their next meal. They will even stake out an area with a good food source and come back to it when they know they're least likely to run into trouble . Other times they don't care and can be unbelievable bold. This had to take more skill than one pudgy old dog acting alone...imho.