My thought then runs to....if they are 10 to 12 weeks of age, and off heat, they should be out in the yard and not in your bathroom.
Exercise also prevents constipation. At 10 to 12 weeks of age they need fresh air and lots of dirt to scratch in.
As to your anatomical question, actually yes, the hen's egg duct and bowel are connected to or part of the same tract. The egg tract joins into the main tract which passes the fecal material. When laying, the egg tract inverts and extends to allow an egg to pass through the main tract without touching fecal material. However, with a large egg, fecal material may also be pushed through as well.
A hen will lay generally by standing up and pushing in the same method that she passes feces. It is the same general motion and stance.
However, 10 to 12 weeks is too young to be laying. It is likely constipation and that is cured by good old fashioned good food, water, and exercise.
LofMc