Being in such a paralytic state from the mass consumption of alcohol that when you reach the toilet you are usually slumped down hugging the bowl and screaming 'Oh God!' into the shit abyss in between violent bouts of upchuck.
A phrase used to ridicule someone's stupidity, "big white mint" refers to a urinal cake. Thus "don't eat the big white mint" implies that someone is stupid enough to mistake the urinal cake for a mint and eat it.