why can't I divide by |y| ?

Yes, you are right. Division by 0 is undefined. But its relevance in this question is very limited since statement 2 states that y > 0. You know that the two statements never conflict so you don't really need to worry about y = 0.

Even if this condition were not mentioned, instead of wondering whether x*|y| > y^2, I would find it far easier to work with x > y*y/|y| since the variables are separated here. I can separately worry about the condition "y=0" if it is possible in a particular question.

My question will be: Is \(x > y*y/|y|\)?

I will think, "y/|y| can be 1 or -1 (assuming y is not 0)"
Statement 1 says x > y. Then can I say that x > -y too (in case y/|y| = -1)? Not necessarily. If y is negative, it may not hold.
e.g. x = 3, y = -4
y/|y| = -1
x is greater than y but x (=3) is not greater than y*y/|y| (=4)
I don't really need to worry about y = 0 here.

Statement 2 clarifies that y is positive. Now we know that y/|y| = 1
Since statement 1 says x > y, it is sufficient to say x > y*y/|y|.

great thanks guys!