Good question... The reason is two fold. A person squares tires to improve handling and I want to begin doing track work (HPDS) in the summer, thus squaring my summer tires. The other is that I already have staggered wheels with fairly new and good all weather tires and can't afford throwing those away to buy two new sets of wheels and tires right now.
Why would you prefer squaring winter tires over summer tires?
The reason I would prefer to stagger summer tires as opposed to winter tires is that with the fat tires on the back there is limited space for chains, if you use them. Also, the fatter tires would be more prone to hydroplaning and other nasty events such as that.
I thought the whole reason to stagger was to improve handling, and if that is true then you actually would benefit from staggered summer tires when you can actually go fast enough to enjoy the added grip.
Leave the all weather tires on the car until they wear out, and get snow tires for winter. Then when the all weather tires (that are really not worth a cr-p for either summer or winter) are worn out, get some real performance tires that have plenty of grip. I'm not meaning to dis all weather tires, I have them on my car. But, I mostly have summer at my house and never have winter, so the rain-management features are a benefit to me, and I don't drive agressively enough to care that my tires are cr-p. I have BFG gForce Radials with the ZR rating. They work for the kind of driving I do, but I'm sure that they are not the best for somebody that actually cares about this sort of thing.
So, keep the tires that are on the car now, and buy a set of tires for winter. Then, next spring put the staggered tires back on for summer and run them until they are worn out. This assumes that staggered tires actually improve the handling...