Usually dogs should have a daily 80% meat and 20% veg if you get down to it.
How do you plan to sustain yourself? Let's start there. Depends on your own variety. How many chickens/ducks? (Chickens that can forage for themselves, good mom's, etc keep the population stable) that's some daily eggs. All the roosters that hatch, you keep for half a year than slaughter. That's some meat. (Keep your mom's safe.)
Fishing could be important. Depending on where you life but go for passive traps if it's smaller rivers. Omega 3 is good for your dogs coat and many other good things but also you.
When hunting, you can easily make chew sticks out of everything you don't want or can't use right now. Sinew, cut out bones, ears, they'll be fine with a day or two of nothing but that. Just plenty of water.
Dogs can handle a day or two without food, that's considered normal in wild populations. They do need protein eventually though, in whatever form it comes. Eggs, meat, insects, the right beans..
Honestly, unless you're feeding your dog after you threw in a stick of butter, most things even if they might need preparation (potatoe, most nightshade) you can seperate before— dogs will be just fine. Just remember protein.
Now cats though..