Oh, those magic numbers. I’ve seen anything from 1 square feet per chicken in the coop all the way to a minimum of 15 square feet per chicken, usually with no mention of run space. Even the commercial chicken industry doesn’t go below 2 square feet per chicken and we don’t normally treat our chickens that way. Four square feet per chicken seems to be the most popular on here but you can see a lot of other opinions.
I personally believe the more space you can give them, within reason, the better. That’s not just for the chickens, but for you. You can follow the link in my signature as to why I think that way.
Can you keep 16 chickens in a 7x7 coop in a Minnesota winter, or spring, summer, or fall for that matter? Probably, if you do some other things. With them in that small area you will probably be working harder at poop management than I like to, but it can be done. If they are locked in that area when they are awake your chances of seeing behavioral problems go up quite a bit. I think those are likely to be your two biggest issues.
Living in Minnesota you are probably aware that snow doesn’t always fall straight down. Covering part of the run will help, but snow blows in from the side, either from drifting or while it is still falling. My chickens don’t like a cold wind in winter either. Summers are different, warm winds don’t bother them, but if a cold wind is blowing they will not be out in it. You can greatly increase the available room in winter by not only covering part of the run but by blocking snow and wind from coming in from the side. Your concern there is not just snow load on the top but wind load on a side. Even if you blow snow out of there to give them space, that’s an example of working harder and not necessarily at your convenience. You may still need wind blocks down low. Another issue with that is that it needs to be available when they are awake. That space in the run doesn’t do any good if they are locked in the coop only while you sleep in on a Saturday morning.
There are differences in what you can do and maybe get away with it versus what I’d want to do. I just don’t like working that hard or living my life on their schedule more than mine.
Now to your basic question, how big a run do you need? I don’t have a set answer for that either, other than the more the better like coop space. If they are all the same age and sex, not nearly as much as if you let a broody hen raise chicks with the flock or you integrate chickens at some time in the future. The main idea with room as far as chicken behavior is concerned is that they have room to get away from each other when there is conflict. Having separate coop space and run space enables a chicken to go where the bully is not. Having enough room so the chicken doesn’t get trapped in the coop or run is another plus (the more space the better), but just having separate places available is a big step up.
I wish I had a good number to give you, I don’t. Look at your budget, your run space and how it is built, and then provide as much truly sheltered space as you can and see how it goes.