The Beguine movement was a free association of women who shared a mystical spirituality. It flourished for over a hundred years across the german-speaking world from the late 12thC to the degree that every town in the Rhine valley was said to have more than one ‘beguinage’. The male equivalents were called Beghards.
It seems to have arisen from the ministry of Lambert le bègue (the stutterer), a priest in Liège who founded a refuge for destitute widows and orphans of Crusaders. Beguines lived in independent cooperatively-run communities where, unlike the church-controled orders, ‘they took no vow of poverty that they might better relieve the poor, no vow of chastity that they might better remain true to love, and no vow of humility for they recognised noone at their inferior.’
Of course it was too good to last, and in 1311 the Catholic Inquisition, fresh from its successes in massacring the Cathars, Waldensians and other enlightened heretics, condemned the Beguines. After some of its prominent women were burned at the stake the movement ebbed away.
Surprisingly however, the Beguine movement resurfaced in the protestant world 300 years later: and as late as the early 20thC over 1000 women lived in beguinages in the Low Countries, the last of whom died only in 1998.
There appears to be no connection with the Cole Porter song, which refers to a dance from Martinique popularised in the USA in the 1930s!