According to Wiktionary
An allusion to the best and costliest cuts of meat from a hog, considered to be parts above the belly such as the loin, rather than lower parts such as the feet, knuckles, hocks, belly, and jowls.
US, late 1800s; popularized 1940s. The variant forms – live/eat and on/off – are attested since at least the 1930s.
However decades earlier is the phrase on the hog, originally on the hogtrain meaning someone living on little expense.
According to Merriam-Webster
High on the hog is an idiom
that refers to a type of lifestyle that reflects
: in a luxurious style. Those movie stars live pretty high on the hog.