This phrase has entered the lexicon from the classic story of the Three Little Pigs:

Wolf: "Little pig, little pig, let me in!"
Pig: "Not by the hair of my chinny chin chin!"
"The hair of my chin" is the beard, for a long time an important symbol of maleness in Christianity, and still one in (eg.) Islam and Sikhism. The phrase "by my beard" was a mild oath used by Shakespeare among others (cf. The Two Gentlemen of Verona, Act 4, Scene 1), implying that the speaker would rather emasculate his chin than fail to live up to his promise. Here "hair of my chin" is used because it rhymes with "in", and padded with three extra syllables to make it fit the meter.

The odd part is that pigs do not have much in the way of a beard, but perhaps that only serves to make the rhyme more memorable. Another possibility is that the pig was speaking Japanese, in which case the phrase takes on a rather different meaning...

A nodeshell rescue! w00t!