Dogs don't really need much, since they are closer to being wild animals than we are. The fur coat really helps. I bring a little mini-tent as a portable kennel, just because she feels more secure when she's got a "home". She sleeps in the big tent with us at night though. I also have half of a yoga mat that I lay out so she doesn't have to lay in the dirt, and I also use one in the big tent by the door to keep her from tracking in dirt. If it's going to be at all warm out, I bring a cool-pad: a water filled pad that helps keep the dog cool. Dog's have much more problems keeping cool than warm.

Most of a dog's first aid needs can be met out of the human kit.