Wow. I've never had to think about this. Basically, it means "not even by the tiniest possibility". Like a hair on the piggy's chin. If you're wondering about "chinny chin chin", it's for rhythm. Little pig, little pig, let me come in! Not by the hair on my chinny chin chin! Then I'll huff and I'll puff and I'll blow your house in!

By the beard of Zeus!

Actually, men used to make oaths and swear by their beard.