When you say "5 square miles" you are describing the area not the shape. A piece of land that is 1 mile wide and 5 miles long or a piece of land that is 10 miles long and half a mile wide both have an area of 5 square miles. Also a circular piece of land that has a radius of 1.26 miles has an area of approximately 5 square miles.
On the other hand the term "a 5-mile square" to me says a square plot of land that measures 5 miles on each side. This plot would have an area of 5 5 = 25 square miles.
The Wikipedia encyclopedia says that "A township in the United States refers to a small geographic area, ranging in size from 6 to 54 square miles ..." and hence I think that the person who correct you is closer to the truth. I doubt however that the townships you mention were all square but to be sure you would ned to check the historical record. Without knowing the details I think you would be better to say "when the Connecticut Land Company sold the Western Reserve land, 25 square miles constituted a township".
I hope this helps,