The rules don't strictly deny square images (here's one). And some portrait images have been cropped (example) or rotated (example) into landscape images. Until August 2011, every entry had to be cropped somehow, since no camera outputs in aspect ratio of 375×210 px - and there weren't less entries than now. So the "price we pay" might be lower than you think.
The contest rules are quite reasonable given that the whole idea of the contest is to select a header image. The personal entry barrier for any author is pretty low - just being an active member of community (ie. having enough reputation to access meta and post images) is enough. It was never meant to be for showcasing all photographic work (e.g. panoramas, timelapses, 3D and erotic images are also ruled out).
Oh, and actually, there is a special contest for portrait-orientation images - the Community Ads campaign. Link your image to a relevant question or some other photography-related page, embed a few catchy words and five upvotes will grant your image randomly (among other qualifying entries) placement for much longer than a week - until the end of year.