Thanks for your responses. I am looking into getting a camera that faces back there. It's expensive but so is an autopsy we just don't have a lot of money.
I think they're poisoning them because the dog can no longer get through without some direct help from the neighbors. We have wire burried along the fence so it would have to dig 1 foot under ground and 2 feet over... not going to happen. So I think they've gone to plan B. I hope it's just me being paranoid, but with neighbors that will look you in the face and lie to you - while you're holding a beautiful and dying hen that's their fault - well they have no soul. I talked to the neighbors on the opposite side of them and they say the guy is nice but his wife will lie to your face even when you have pictures proving she's lying.
We have a camera in our coop (see sig), and it's really not difficult to set up, if you have electricity and wireless internet within range. Most relatively inexpensive internet cameras have motion detection to only record when there's a (potential) intruder. The trick would be to aim the camera higher than the chickens (so it doesn't record all of the time), and low enoughto catch the backsideholes.