One of the great benefits of the square is that it engages the viewer differently. Instead of creating a natural side-to-side or top-to-bottom flow across a frame, the square tends to make people move around the frame. Elements of particular interest or emphasis in the frame will create pauses or anchor points, but your eye will quickly move on from one of those anchor points and resume its scan around the image in a circular fashion. The square format also lends itself to images with simple, graphic shapes.
One could crop a 16:9, 4:3 or 3:2 image into a square, but cropping isn’t the same as the experience of seeing square and composing a square image directly in the camera. Square image files are smaller, but the "pixels-per-inch" resolution is the same, even though the captured image size is smaller. For example, the Panasonic Lumix DMC-G6 is capable of producing 4608×3456-pixel images (16 megapixels), but when shooting 1:1, the dimensions change to 3456×3456 (12 megapixels).
Because square images ignore the extreme corners of the sensor, areas where lens performance is almost always the worst, 1:1 shots are generally sharper. And by using only the sweet spot in the middle of the lens, vignetting is nonexistent. This unexpected benefit makes it more practical to use polarizers and other filters that could otherwise darken the corners of the frame.
Try shooting square. It’s more than just putting blinders on your camera. It’s a whole new way to examine your little quadrilateral corner of the world.