This is part of Kent's bitter denunciation of toadies like Oswald, who always say whatever they believe the boss [master] wants to hear. Kent says the toadies says no, yes ["renege, affirm"], and alter their facial expression ["halcyon beaks"weathervane faces] with every shifting wind of emotion ["every gale and vary"].
Kingfisher genus: Halcyon Alcyone and Ceyx Halcyon beaks is a phrase with an interesting background: in Greek fable the halcyon [a kingfisher] was a bird that was supposed to have the power to calm the wind and the waves while it nested on the sea during the winter solstice. Alcyone or alkyon [derived from the Greek word alkyon meaning "kingfisher"] was the daughter of Aeolus, who married Ceyx, son of Eosphorus the Morning Star. Alcyone and Ceyx were very happy together in Trachis, and according to some accounts, often sacrilegiously called each other "Zeus" and "Hera." This angered the god Zeus, so while Ceyx was at sea [going to consult an oracle according to Ovid's account], Zeus threw a thunderbolt at Ceyx's ship. Soon after, Morpheus [god of dreams] disguised as Ceyx, Alcyone's husband appeared as an apparition to tell Alcyone of his fate, and out of grief, she threw herself into the sea. Out of compassion, the gods changed them both into halcyon birds, named after Alcyone.