From what I've been reading today, the "penguin" walk is more likely when they are egg bound, and the "duck" walk is more likely when they have a belly full of fluid or "cooked" egg yolks that have gotten into the abdomen (internal laying). I hate to think of her being in pain. I just don't have any good ideas about what to do for her. With all her egg issues this does make sense, and at least if that is the problem it won't affect the other hens, except to the degree that it might be genetic and the other black star is a hatch mate. I'm at a loss, but can find no remedy for this condition. It just seems like a lot of people have been having this problem. I know it doesn't explain your roosters though. How are they doing?
It was just one rooster who had the circling problem. As for the staggering /sideways walking, I'm pretty sure it has nothing to do with egg laying in my hens. In the two recent ones who had this problem, it began immediately after they went into a heavy molt, and they had already stopped laying for the season. They did have that duck walk kind of thing, but that was in association with staggering, which really didn't look like an uncomfortable abdomen kind of thing. It looked like a balance or neuromuscular problem to me. Their new feathers are opening up now, and they are both walking normally.
I had one other hen this past summer with the staggering problem. This one was not molting at the time, and continued to lay big, beautiful eggs throughout the whole time, so I don't think it was a laying problem. She mostly recovered, except that she occasionally takes a backwards or sideways step. She had only a very light molt this fall, and she continued to walk pretty normally throughout the molt.
For mine, I feel pretty sure that they are carriers of whatever disease afflicted the rooster, but most of my flock are resistant to it, with the occasional bird showing symptoms, especially if stressed (as when molting). I have 16 birds at the moment, and most of them have never had any of these symptoms, so clearly some birds are resistant.
I've had several different breeds, but never a black star, so I am not familiar with their molting and laying patterns and problems.