Does DayZ run on linux?
DayZ is a mod for ArmA II, which is a Windows only game. You must own and your computer must be capable of playing ArmA II (specifically, its expansion ArmA II: Combined Operations, which includes the main game and a later expansion) in order to play DayZ. So, unfortunately, it cannot be played on a Linux box.
If you are asking if you can run a private hive DayZ server on Linux, then the answer is yes. You have to run it with wine, but it can be done. You cannot however run the game client on Linux.