A pickleback is a shot of whiskey followed by a shot of brine from a pickle.
When Archer rants about the "stupid logic puzzle", he says, "Who am I, Bernard Bolzano?" Bolzano was a Bohemian mathematician and philosopher devoted to the a priori sciences of logic, mathematics, and theology.
When they first arrive on Antarctica, the gang jokingly discusses what may have happened at the research station that led to the death of one of the people there. They come up with five scenarios - a thawed alien monstrosity (reference to John Carpenter's The Thing (1982)), demon possession (a reference to Sometimes They Come Back... for More (1998), aka Ice Station Erebus, the in-name-only final entry in a series of three films based on Stephen King's short story of the same name), "sex worms" (a reference to David Cronenberg's horror film Shivers (1975)), flesh-eating virus (a reference to the Cabin Fever (2002) film franchise), and "a multidimensional hell-beast" which will make people change their perspective on reality and existence (a direct reference to H.P. Lovecraft's novella The Mountains of Madness, part of the Cthulhu Mythos).
There is a poster for a Rona Thorne movie called Consumed in the Antarctic base.