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).
In this episode McMurdo Station is made out to be a privately owned and funded research outpost. In real life McMurdo Station is operated through the United States Antarctic Program, which is part of the government funded National Science Foundation. McMurdo is the largest settlement in Antarctica, with the capability of supporting up to 1,258 residents and is one of only three US-operated science facilities to have residents year-round. It is the location of the only airport on the continent that operates year round, most supplies flown or shipped to Antarctica pass through McMurdo Station. McMurdo is also the location of a three-bed hospital (the only one on the continent) and has a year-round medical staff. While other outposts in Antarctica have doctors stationed at them, most are not there year round, and they are not equipped to deal with medical emergencies.