The natural numbers extend infinitely in the positive direction (this may not be the proper terminology), so for every $x\in \mathbb{N}$ there is a $y \in \mathbb{N}$ (namely, $x+1$) such that $x\leq y$ isn't there? Yet my textbook says that this statement is false...why?
Your textbook made an error. Your reasoning is correct. Textbooks do make errors sometimes. As suggested in a comment, the textbook most like meant to say that the following statement is false: $\exists y\in\mathbb{N},\forall x\in\mathbb{N}\; (x\leq y)$