Some updates. I apologize ahead of time for how long this is, but I am throwing out every detail I can think of. She has pooped several times. It isn't well formed, but there is a fair amount of it.
I offered her some electro/vit/probiotic water and she drank a good bit. So, I figured I would offer her some food and she was quite hungry. She ate several tablespoons of purina layer pellets, with a little corn mixed in (what the others are eating right now). I also gave her some dried meal worms and she scarfed those down as well. She would eat for a minute or two, take a few minute break, eat again, repeat. She refuses to bend over to get to anything. I made a makeshift shelf from a box that was a little below beak height and she was happy with that. After eating, she pooped and is sleeping again, in my bathroom.
I examined the fresh poop under a microscope and really don't see anything unusual at all.
I palpated her, by putting a gloved and lubricated finger into her vent, but couldn't feel anything but soft tissue and bone. Also, I tried exploring more upward and in, but the duct seemed to direct my finger more downward. At the same time, I felt around her abdomen and didn't feel anything that seemed unusual. She accepted me doing this exam, but was not happy about it.
Anyway, I took a few more pics, in case they might be helpful. She almost makes me think of an owl, the way she has been standing.
I have to wonder if the other chickens just kept her away from things until she got sick. The chickens are truly free range, able to roam anywhere over several acres that are mine or my neighbors' yards, so I can't always tell where they are or what they are doing/getting into. I can tell you that the other three hens seem to shun her and normally she will not let anyone touch her. She has always been skittish. I am still learning and hope that I didn't make some sort of giant mistake by allowing them to behave naturally.
I work with dogs that have behavior problems, on a volunteer basis, and usually take on pretty severe cases. So, I am comfortable with dogs and I would never just allow them to determine their "biting" order. Maybe I shouldn't have done that with the chickens? The dogs I rehabilitated and kept have all killed before I rehabilitated them, but now they are wonderful. My lab and husky will take turns playing chase with the chickens, sometimes letting the chickens chase them, and many nights I will look on the porch and the chickens are bedded down, laying against one of the dogs. We have many predators in this area and the dogs do an excellent job of protecting the chickens. My cats are afraid of the chickens.
As far as egg production goes, I haven't seen eggs from any of them in over a month. However, because of their ability to roam, that is not unusual. They make a nest somewhere, lay eggs in it, and I have to search for it. After I find it and take the eggs, they sometimes move to a new location. It is kind of like an easter egg hunt all year long. I guess there may be some down sides to this, if eggs are not found and start inviting more predators or if the chickens consume a bad egg. I was thinking of locking them in their coop at night and letting them out mid-morning, as a friend said they will be more likely to consolidate their eggs in the coop that way.
Also, since they roam, they sometimes will eat from my compost pile, which includes used coffee grounds. Short of fencing around the compost pile, I'm not really sure if or how to keep them out of it. Maybe that has something to do with it?
Also, when I tried to post pics with my original post, it was from my phone and I was using the mobile version of the site. When I switched to the full version from my phone, I was able to post pictures. Not sure who I should report that bug to.