The easiest way to go about this question is to sketch it and identify which are the diagonally opposite vertices. However there is a shorter way to accomplish the same.

We know that diagonals of a square bisect each other. So in a square ABCD, the coordinates of the point where the diagonals of ABCD intersect.

$\frac{A + C}{2} = \frac{B + D}{2}$

or coordinates of the point where the diagonal intersects can be written as = $\frac{A+B+C+D}{4}$.

So $x= \frac{-1 -1 +2 +2}{4}=\frac{1}{2}$
$y= \frac{-2 +1 +1 -2}{4}=\frac{-1}{2}$.

Hence option B is correct!