According to the OED, the term is borrowed from the Banyans in the East Indies, a caste that eat nothing that had life.
banyan day (plural banyan days)
- (dated, Britain, nautical, idiomatic) In British naval tradition, a day of the week when galley kitchens served no meat on board ship.
- (Britain, nautical, idiomatic) A picnic or cookout for the ship's crew.
- The Dictionary of the Vulgar Tongue / Lexicon Balatronicum: A Dictionary of Buckish Slang, University Wit, and Pickpocket Eloquence: altered and enlarged (London; 1811)