This article is from the Hiking/Backpacking With Canines, by Terri Watson with numerous contributions by others.
You should not put anything in the dog's pack that you cannot afford
to lose. Fragile items, or items that should not get wet, are also a
poor choice. The dog will often scramble though tight spots with less
grace than his human counterparts, banging the pack against rocks or
trees in the process. This is particularly true with a novice dog that
has still not learned how much wider they are with a filled pack on.
Another must is to pack both sides with roughly equal weight and
volume. Failure to do this will result in, at best, a dog that's
off-balance, and at worst, one side of the pack flipping over to the
other, or the whole pack sliding over to one side.
Items that I have put at various times in the dog's pack: extra water,
rain gear, 3/4 length sleeping pad, dog food (sealed in plastic bags),
dog comb, dog frisbee, extra bags for cleanup, the camp stove inside
of cooking pots (so it was protected from banging), camp soap and