Definition: The word "square," when used in contest with roofing (shingles) and exterior siding, means one-hundred square feet. So, a square of shingles is 100 square feet of shingles. A square of siding is 100 square feet of siding.
One hundred square feet is ten feet by ten feet.
One reason why roofers and siders speak in terms of squares is because they deal with such vast quantities of materials, that it is pointless to discuss smaller numbers. For example, shingles come in bundles. Three or four bundles of shingles typically comprise a square.
"The contractor estimated that 23 squares of shingles would be sufficient to cover the 2,234 square feet of roof area."