I'm yet to encounter a dog who likes mushrooms but the assortment of working sheepdogs, JRTs, Labs and Scotties, plus a Cairn and a Basset that I've known says that pretty much anything else goes.
Not saying I'd ever put my dog on a vegetarian diet but they can consume non-meat products, which technically makes them omnivores rather than exclusive carnivores. As I understand it, cats shouldn't be fed anything but a meat-based diet.
FWIW, my vet recommends mashed potatoes and scrambled eggs (no dairy in either) as a recovery meal for dogs with dodgy bellies.