For example, person A states something. Person B says "And pigs fly" to imply person A was wrong.
If there's no term for it, what could you call that that sounds smart?
This is diasyrmus, which, according to the Silva Rhetoricae definition is "[r]ejecting an argument through ridiculous comparison."
In this case the comparison is implied: person A's statement is as false or absurd as the notion of flying pigs.
To a logician, your examples would be reductio ad absurdum. "If that's true, then I'm a Dutchman/the Queen" is both a well-attested phrase and a valid proposition, implying that whatever it is is false. Similarly, "And pigs fly" asserts the equivalency of what has just been said to a clearly absurd proposition. (Martin Gardner, in The Annotated Snark, traces the phrase to "an old Scottish saying, 'Pigs may fly, but it isn't likely'.")
A general word for such a response is retort, before which you can use a relevant adjective if necessary.
You could call the above, for instance, a sarcastic or witty retort. (Though I think you'd have to do a bit better than "and pigs fly" to merit the word witty.)
Another option is riposte.
I should add that neither retorts nor ripostes need be "false" themselves. They do, however, highlight the falsity or absurdity of what was just said.