"Root hog or die" is a common American catch-phrase dating at least to the early 1800s. Coming from the early colonial practice of turning pigs loose in the woods to fend for themselves, the term is an idiomatic expression for self-reliance.
The word "root" is used as an imperative verb, as is "die". The phrase (sometimes rendered with commas after root and hog) can be parsed as, "Root (search for roots to eat), hog, or [you will] die."