Well, better not!
Fungi/mushrooms inhabit soils throughout the world. Some inhabit the soil of the Mississippi Valley, the Middle Atlantic States, and southern Canada; others grow in soil contaminated with bird or bat excrement.
Active dogs who spend a lot of time running outdoors (like the hunting dog), especially dogs who love to dig, risk becoming infected with fungal organisms that live in the soil. Usually a strong immune system will keep these bad bugs at bay. But if the dog is battling another infection or has a weak immune system, these organisms can get a foothold in the dog’s body and cause serious disease.
These fungal agents cause arthritis, pneumonia, infections in bones, and signs of systemic infection such as fever, loss of appetite, and malaise. No vaccines are available for these organisms. If you live in an area where fungi are present, the best way to prevent infection is to keep your dog from digging, especially around the holes of burrowing animals where the fungi are especially abundant.
Never allow your dog to eat wild mushrooms!
Theoretically, mushrooms from the supermarket should be okay.