The answer is Yes and No, the Players on the Ice need to wear one. Goalies don't have to as the "NHL Rules" only said that "Goaltenders are highly suggested to wear a sort of protective helmet" which means they don't have to if they don't want to. But if they come into the game with one they must wear it during the game but if they show up without one no one can stop him from playing because of it.
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No, full face masks or cages are not optional in the NHL due to rule 9.6 Dangerous Equipment, "A mask or protector of a design approved by the League may be worn by a player who has sustained a facial injury."
In other words you cant
where the full mask unless you have an injury, otherwise you are only aloud to where a half visor.