If you are used to backpacking with a dog, the following might be old news, but here is what has worked for me in conditions like the Canine John Muir.
You'll want an assist handle. You may encounter fallen logs (somewhat rarely on this trail), but stream crossings are frequent (see the conditions page). Put items in the dog pack which can get wet.
Whether your dog needs to wear booties depends on how tough your dog's pads are, how much distance you cover each day, and the terrain. The trails of the John Muir aren't particularly hard on a dog's pads, but there certainly are stretches of walking on rock or gravel. I've also successfully used booties for hot sand, but be careful here: the pads are one of the places your dog sheds heat, so watch for overheating. Although I rarely put on my dog's booties, I do take along a set, in case of injury or sore paws.
Some dogs wear booties in snow to prevent ice from forming between the toes. This is most likely to make sense in winter and on relatively level ground. The soft snow of the John Muir in spring and summer is not as likely to bother your dog's feet, and the steep slopes will require more traction than booties can provide.
You need to prepare for a wide range of temperatures. Afternoon temperatures will often be hot and nighttime temperatures may go below freezing at higher elevations even in the summer. Shorter haired dogs will need help in keeping warm at night. A jacket like the Ruff Wear K-9 Overcoat II is more insulated than the doggie sweaters which are sold for non-backpacking purposes. Unlike a blanket or quilt, you don't need to worry about it falling off during the night as your dog moves around. Having your dog sleep in your tent will help keep them warm, and also give you the chance to notice if they are shivering. A sleeping pad is just as important as a jacket/blanket/quilt in keeping warm. Any of the closed cell foam sleeping pads sold for humans can be cut down to dog size (or kept as-is, which reduces the need to convince your dog to lie down in just the right spot).
The streams and lakes along the John Muir contain pathogens such as Giardia or E. Coli, although the quantities (to the extent that we have test results) are low enough that some backpackers choose to drink unfiltered water. The issue is particularly relevant for dogs: you are unlikely to prevent your dog from drinking from streams or puddles (and if you are tempted to try, make sure to offer water to your dog water frequently; dehydration is a greater health risk in this area than water-borne illness).
As with human food, you want to reduce weight, while getting enough calories and nutrition. That means high fat, high protein, moderate carbohydrates, and as little water as possible. We generally pack some food which is suitable for either humans or dogs, such as jerky, summer sausage, cheese, and the like. But our dog's main food on the trail is a dry dog food, Eukanuba Premium Performance (I see Eukanuba also makes a Maximum-Calorie food; haven't tried that one for backpacking). Another high-calorie food is Science Diet Active Adult. Puppy food is also somewhat higher-calorie than adult dog food. Avoid low-calorie foods such as those sold for overweight dogs. And of course canned dog food is about 80% water and should be avoided whenever possible. You can compare calorie density on the web sites of the manufacturers, or on the label (if it doesn't give calories per kilogram of food, or cup of food, look for high percentages of fat and protein, which are found on all dog food labels).
I recommend trying out your backpacking food at home at some point. The trail would be a bad place to find out that it makes your dog sick.
Camping With Your Dog(s) - aimed at the car camper, but worth reading.
Canine Activities: Camping and Backpacking FAQ
Ruff Wear - quality gear.
Wolf Packs - haven't tried their products, but they seem to be well-thought-out designs. The web site also has a lot of general information about backpacking with a dog.
Eukanuba and Science Diet make high-calorie-density dog foods. Both web sites will tell you the calorie density (kilocalories per cup).
Giardia Lamblia and Giardiasis With Particular Attention to the Sierra Nevada, by Robert L. Rockwell, PhD, Yosemite Association News Letter, #4, March 18, 2002 (web article updated March 19, 2002). A good paper, although it only covers Giardia.
Coliform and Pathologic Bacteria in Sierra Nevada National Forest Wilderness Area Lakes and Streams, Robert W. Derlet, MD; James R. Carlson, PhD; Mikla N. Noponen, MD, Wilderness and Environmental Medicine: Vol. 15, No. 4, pp. 245–249 (2004). Or read a news article based on the paper. Measures water pathogens in 100 backcountry lakes and streams in 2003.