square dance vs square dancing

square dance and square dancing both are nouns.

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

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

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

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

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