Any number times itself is a positive number (or zero), so you can't ever get to a negative number by squaring. Since square roots undo squaring, negative numbers can't have square roots.
Yes, you can square a negative number. In fact, any number at all can be squared, even numbers like pi and 0. This is because to square a number just means to multiply it by itself. For example, (-2) squared is (-2) (-2) = 4. Note that this is positive because when you multiply two negative numbers you get a positive result.
