I am one of those who don't actually like the Accuquilt strip dies. I always have to stop halfway through and straighten the fabric; otherwise I get a wrinkle (and miscut) in the middle of the cut. I have not tried cutting strips into squares using the strip die, as I'm pretty sure mine would not come out as accurate as I would like.
Have you tried cutting strips with a June Tailor Shape Cut Pro (or the newer Stripology ruler which has been the subject of some recent threads)? For me, it's much easier to cut accurate squares using one of these ruler; I just align the strip with the markings on the ruler, then cut through the slots.