The title refers to The Eiger Sanction (1975) staring Clint Eastwood. Ironically whilst the film is serious it is based on a book intended as a spoof of Ian Fleming's James Bond novels. This episode is therefore a spoof of James Bond spoofing a film based on a book spoofing James Bond.
The eye that pops up in the hot tub is a reference to the trash compactor in Star Wars: Episode IV - A New Hope (1977).
Charles Fredric Andrus (the name Cheryl drops) was a horticulturist who worked to make disease resistant vegetables. He invented the first anthracnose resistance watermelon named "Congo".
The drink they were drinking the night before was Glühwein, a hot beverage usually made with red wine along with various mulling spices and sometimes raisins.
When Archer says, "PM Dawn" and Ray says, "Set adrift on memory bliss", it's a reference to the musical group PM Dawn's song "Set adrift on memory bliss".