square dancing vs square dance

square dancing and square dance both are nouns.

Word	Noun	Adjective	Verb	Adverb
square dancing	Yes	No	No	No
square dance	Yes	No	Yes	No

As nouns, square dancing and square dance are synonyms defined as:

square dancing and square dance: American country dancing in which couples form squares

square dancing (noun)	square dance (noun)
American country dancing in which couples form squares	American country dancing in which couples form squares

square dancing (verb)	square dance (verb)
	dance in formation