So I have done a similar tutorial and a written pattern few years ago but it was only using double crochet for the square, this is the classic one though which everyone uses and looks nice as well.
I love when it's made with cotton as you can created many projects using this pattern, be it a pillow cover, a summer throw or a rectangular shawl, many ideas come to my mind.
For the other version of how to turn a circle into a square, you can find it here (Video)
ch4, 10 dc in the first st from the hook, join.
Round 1: ch2, 2dc in every st, join.
Round 2: ch2, 1 dc in the first st, 2dc in the next st, alternate till end of the row.
Row 1: sl st in any st, starting with the corner, ch3, 1dc in the same st, ch2, 2dc in the same st ( 1st corner created), 1hdc in the next 2 st, sc in the next 3 st, hdc in the next 2 st, corner again, repeat till end of the row, after the last hdc you might have an extra st, skip and join to the beginning ch3.
Row 2 : ch3, in the corner, 2dc in the sp, ch2, 2dc in the same sp, 1 dc in every st till the next corner, repeat, join at the beginning ch 3. Fasten off.