When you have more than a pair in one cage, they should never have access to anything they can use as a nest. One of your female decided the little box was a nest and laid eggs. Since none of them have sat on the eggs, I would discard the eggs and the box. Moving them wouldn’t work, even if you knew which bird laid them, because once eggs are relocated she would ignore the eggs. Unless you have observed a male and female showing signs of being bonded – sitting together, feeding and grooming each other, chasing the other two birds away, mating – then I doubt the eggs are fertile anyway. It just sounds like the box stimulated one of the hens to lay eggs.
If you wanted to breed your birds, you would have to separate the pair to another cage and probably another room and get them a nest box. There is a lot more involved which I can help you with if you decide to go that route.
If you want the four birds to remain together peacefully, and not have more eggs, then you need to make some changes. It is not good for a female to lay eggs if she isn’t with a male and set up with a nest box. It’s just a lot of unnecessary strain on her health. And without making changes, she might keep laying eggs, which can eventually kill her.
You need to do all of these things to discourage egg laying. Keep in mind that to lay eggs, she needs longer daylight, warmer weather, abundant food, and a quiet, private environment. Your goal is to reverse these conditions.
Limit her light to 8-10 hours by covering the cage early each evening
Do not give her anything to use as a nest – no bird huts or tents, no box, bowl, etc. If she decides to sit in a food bowl, remove it and replace with smaller cups.
Do not give her anything to shred such as paper or cardboard.
Rearrange the toys in the cage frequently.
Move the cage to a different place in the room. Move the cage about once a week, or whenever she shows signs of nesting – settling on the cage floor for example.
If she is let out of the cage, do not let her get in any dark cozy places and don’t give her free roam.
When you handle her, limit any petting to her head and neck – do not pet her on the body.
If there is no metal floor grate, then do not use any bedding or paper in the cage tray – leave it bare and clean it daily.
Thank you for asking Lafeber,