Well I watched it about 2 dozen times and I'm fairly convinced it was already dead.
Granted, this video is still in very poor taste.
It is fact that AQ and other terrorist types have used dead animals as a hiding location for IEDs before. There are a lot of videos of soldiers shooting and burning dead animals to make certain they aren't armed. (that the left posts under headlines of "US soldier kills and burns dog", etc..) Its terrible, but its true. I suspect this puppy drew concern initially for the same reason, then it became a joke.
Sadly all soldiers aren't innocent. there is a video of a couple of soldiers who toss rocks at an injured dog to get it to walk, they videoed it because this particular dog walks on two legs (the left front and the right rear or vice versa) and they found it amusing.