When we got back to the house, I picked up a dog biscuit to give to her because she'd been a good dog and did her business outside. At first she refused the biscuit, then she very carefully deposited something on the floor. She'd been carrying an egg in her mouth! No wonder she was being so careful.
It's a fairly large egg, slightly smaller than a golf ball, and a light tan color. I'm guessing it's a pheasant, because we have lots of them around here, they nest on the ground, and they're smaller than a chicken but still larger than most of the local birds.
I tried to find the nest to put the egg back, but didn't have any luck. It was in an area with lots of very thick grass and I was afraid I'd just end up disturbing the nest further or even accidentally stepping on it. I also tried calling the vet (closed), the Nebraska Humane Society (they don't deal with wild birds) and a local wildlife rehab center (no answer) to get some advice.
I wonder why the dog was so gentle with the egg. If she thought of it as food, why didn't she just eat it right on the spot? She has exhibited strong mothering tendencies in the past, but an egg is totally different from a puppy.