According to my calculator, -2^2 is equal to -4, I did this to another calculator, and google, but got the same answer.

But, isn't
-2^2
= -2*-2
= 4 according to negative multiplied by negative = positive?
So which one should I follow?

Your calculator is programmed to follow the order of operation.

Since exponents are to be done first, it is doing

-1(2^2)
= -1(4)
= -4

Brackets on your calculator allow you to change BEDMAS
trying entering it as
(-2)^2 and you will get +4

Suppose you were to calculate

5 - 2^2, I am sure you would get 5 - 4 = 1
that is, you would do the power first , then do the subtraction
What the calculator does in effect is
0 - 2^2