Square back indicates there is a board strip in the spine and the book block is not rounded. We have also made books with a square back book block but the cover has a paper backliner if that is the look you are after.
Some books need to be square back due to limitations of our automated equipment. Thin books (under 3/8” in thickness) and oblong books are the two most common applications for a square back book.
Mechanically bound casebound books normally refer to a wire-o bound book block that is glued to a hard cover. These can be both round back (this will have two separate endsheets on thicker cover stock) or square back (this will use a one piece endsheet normally on 15pt cover stock that we score in the middle to create a “V” shape when closed). We have automated processes for both round and square back casebound wire-o books. While not as common, we can also case in a book block that uses a spiral wire or plastic coil binding method.