Solution:
We need to know if lxl > lyl or not.
Consider first (1) alone.
It says x^2 > y^2.
Or (x^2 - y^2) > 0.
Or (x+y)(x-y) > 0.
There can be 2 cases.
Case (1) x+y > 0 and x-y > 0 which is x > -y and x >y.
This means x > 0 and x>lyl.
Or it means lxl > lyl
Case (2) x+y < 0 and x-y < 0 which is x < -y and x < y or -x > y and -x > -y.
This means x < 0 and -x > lyl.
Since lxl = -x if x < 0, we have lxl > lyl.
Since both cases give lxl > lyl, we get (1) alone is sufficient.
Next consider (2) alone.
Let x = 2, y = 1. So lxl = 2 and lyl = 1.
Here x > y and lxl > lyl.
Next let x = -3, y = -4. So lxl = 3 and lyl = 4.
Here x > y but lxl < lyl.
Since nothing definite can be said (2) alone is not sufficient.

The correct answer is (A).

What if there is a statement x^3>y^3, then |x|>|y| is true because the power is odd, right?

Not necessary.
Take same examples.
Let x = 3 and y = 2.
x^3 = 27 and y^3 = 8. lxl = 27 and lyl = 8
Here x^3 > y^3 and lxl > lyl.
Next let x = -3 and y = -4.
So x^3 = -27 and y^3 = -64. lxl = 3 and lyl = 4.
Here x^3 > y^3 but lxl < lyl.
Hence it is not sufficient.

Actually when you are solving Data sufficiency problem, you are not supposed to ascertain whether statements (1) and (2) are correct or not.

You first note what has been asked in the main question?
You then take the statement (1) and try to find out whether you can answer the main question assuming (1) to be true. The same thing is then done for statement (2) and lastly for both statements combined if individually both statements do not give a definite reply.

In your solution you cannot take x = -2 and y = -3 because they are contradicting what is given in the statement (1) which is x^2 > y^2 since (x^2 = 4) < (y^2 = 9)
You need to take x and y such that x^2 > y^2.

Hope it is clearer.