I reallllllllllly don't think the producers "coach" anyone to DO anything on the show. Especially when it can be viewed as a physical threat to another contestant. Smashing an egg on someone's head is a physical threat imho...and believe me, IF Zach WAS coached into doing it...looks to me like he didn't struggle with whether or not he would. He seemed to relish in the other guy's embarrassment.
He got the egg from the refrigerator, probably right next to the beer.
They're going to be living there....they need food. I find it highly unlikely that a producer brought an egg in hopes someone would smash it on someone's head.