questions
She crossed the grass to her car, and laughed when she saw the white square of paper under the windshield.

She crossed the grass to her car, and laughed when she saw the white SQUARE PAPER under the windshield.

Question 1: the white square of paper == the white SQUARE PAPER? If not, what’s the difference between them?

Question 2: the white square of paper --- what’s the meaning of ‘of?’

Question

questions
She crossed the grass to her car, and laughed when she saw the white square of paper under the windshield.

She crossed the grass to her car, and laughed when she saw the white SQUARE PAPER under the windshield.

Question 1: the white square of paper == the white SQUARE PAPER? If not, what’s the difference between them?

Question 2: the white square of paper --- what’s the meaning of ‘of?’

Answer

It could be a white square of cloth or a white square of plastic. But it wasn't. It was a note (evidently) written on paper, so it was a square of paper.
Square paper = paper cut in the shape of a square
Square of paper = something square made (folded) of paper 
"Square paper" doesn't sound good for me, maybe because paper is uncountable, I'd rather say "square sheets of paper"/
But I'm not a native, so...)