Yes, I have heard them used. They are a bit quaint though. Be with child = pregnant Born in a barn = uncultured, impolite. Originally, the phrase was specifically meant for people who would enter a room without closing the door behind them.

Just to add a UK perspective to the answers you've had from two US English speakers: It's more or less the same in British English. 'To be with child' is understood, but not used these days, at least not seriously. It's incredibly coy and euphemistic as a phrase. You might come across it in a text from a previous century, or in a Bible reading in church, but otherwise you'd never hear it in a modern context. 'Were you born in a barn?' is occasionally used in a jokey way, especially by older people, when someone walks into a room without closing the door behind them.

A woman can be "with child", meaning pregnant. It is somewhat old fashioned, but still used occasionally, mostly by older generations. It is much more formal than saying pregnant, but very rarely used. "Were you born in a barn": I have always heard either that, or more commonly "were you RAISED in a barn". This would imply, by asking the rhetorical question, that someone does not have good manners, or that they are out of touch with a popular modern or current concept or event.