On the surface, Gone Home doesn't seem to have much to do with BioShock. However, The Fullbright Company's Steve Gaynor revealed that a very tiny easter egg suggests that it is set in the same universe as Irrational's beloved Bioshock franchise--"in a totally non-litigious way," of course.
The nod takes the form of, funnily enough, a video game. BioShock 2's Minerva's Den DLC adventure was Gaynor also worked on, features an old video game titled "Spitfire". In Gone Home, a Super Nintendo game called "Super Spitfire" can be found while exploring the house.
Super Spitfire's publisher, CMP Interactive, is also a callback to the three initials of Minerva's Den's protagonist Charles Milton Porter, Gaynor revealed on the Tone Control: Conversations With Video Game Developers podcast (via IGN).
During the podcast, Gaynor also went on to allude that System Shock 2 is also set within the same universe as Bioshock (and thus Gone Home as well), but there's no direct connection between Gone Home and that classic. Still, Gaynor has managed to subtly tie together disparate stories into one universe--much like J.J. Abrams has with Slusho.