Of course you are going to get different answers. We keep them in so many different climates, in different set-ups, with different flock make-ups, using different management techniques, having different goals, with different breeds, and who knows what other differences, how could one magic number for space cover everyone? I think that’s what the other posters are saying.
I’m always a proponent of providing as much space as you reasonably can for three reasons. I find the more space I give them the less behavioral problems I’m likely to see, I have to work less hard, and I have more flexibility to handle problems. It doesn’t matter that much if the space is in the coop, coop and run, or if they free range and sleep in trees. What matters is how much space is available when they are awake.
Let’s say you want to keep 9 in that 4’ x 4’ coop. All you use it for is to safely lock them up while they sleep. When they are awake they have access to your ¼ acre. That will work fine as long as you let them out at the crack of dawn each day and they have weather and conditions that they can go outside. If you decide to sleep in until noon on weekends or one day that you have the flu, it might get pretty crowded in there. If you decide to take an overnight trip for work or vacation, I guess you have someone dependable to let them out at the crack of dawn.
I’m guessing you’ll feed and water them outside? I mean, how could you configure that coop with roosts for 9 birds and position the food and water where they won’t poop in it.
If you are counting on them having access every day of the year practically all day long each day to that ¼ acre, how do you handle a predator, say a bobcat, fox, dog, or raccoon, that starts picking them off in the middle of the day. Sometimes those predators can be pretty difficult to manage on a short term basis.
If you decide to integrate new chickens or have a broody hatch and raise chicks, it really helps if they have some extra room.
Those are a few examples where extra room gives you a bit of flexibility.
Can you keep 6 to 9 chickens in that 4x4 coop and using that ¼ acre? Yes, you certainly can with the right management techniques as long as things stay ideal. If you run into a problem, you might get stressed a bit if you shoehorn as many as you possibly can in that coop.
My goals and conditions are quite a bit different from yours so I can’t tell you what I would do in your circumstances. I’d be very uncomfortable with more than 4 to 5 hens in that 4x4 coop unless I had a fairly predator resistant run available where they could be locked up if that full 1/4 acre is not available some days.