Okay... my Ti-89 isn't squaring numbers right.

Hello,
You are not using the right syntax. It has to do with priority of operations. raising a number to a power has priority over the negation sign (the change sign key). Th operations you are performing are treated as follows.
1. The square of 2 is calculated first (4),
2. then the negation (change sign) is performed, giving -4.

To get what you intend, namely the square of a negative number, you should enter the calculation as follows.
(-2)^2.
Since the parentheses have priority over everything else, the negation of 2 is calculated first, giving -2, then the power is performed next, and the result should be 4

If you need convincing, here is a screen capture.



Hope it helps.