'squareness' originates from English, specifically formed from the adjective 'square' plus the suffix '-ness', where 'square' entered English via Old French 'esquerre' (from Vulgar Latin *exquadra) and ultimately from Latin 'quadrus' meaning 'square' (related to 'four'), while the suffix '-ness' denotes 'state or quality'.

'square' changed from Old French 'esquerre' (and Medieval Latin/Vulgar Latin forms like *exquadra) into Middle English forms such as 'squar(e)', and later became the modern English 'square'; adding the native English suffix '-ness' produced 'squareness'.

Initially related to 'four' or 'four-sided' shapes (Latin root), the term developed to mean 'having right angles' or 'resembling a square' and later acquired an abstract social sense of being conventional or dull; today 'squareness' covers both the geometric quality and the figurative sense.