Egg cartons will have some sound absorption effect in mid to high frequencies. However they will not soundproof a room. Absorption within a room and soundproofing to block sound aren't the same thing.
The folks at The Station nightclub in Rhode Island assumed this as well, and the terrible irony is that the cheap foam they used for "soundproofing" their stage from complaining neighbors was as acoustically ineffective as it was dangerous.
Which of course brings up the other obvious problem with egg cartons. They are flammable. Many states (including Rhode Island!) will not let you legally cover walls, commercial or residential, with materials which have anything less than a class "A" (also called class 1) flame-spread rating.
Please, please do not apply coverings to walls which have anything other than a class "A" rating! Many acoustical foams are legitimately sold which are not class "A" rated, because fire codes in various states do not require it. I think that this is a dangerous mistake.