Vic Lynn played for all 6, I think he is the only one:
NYR: 42-43 (1GP)
Detroit: 43-44 (3GP)
Montreal: 45 (2GP)
Toronto: 46-50 (213 reg GP, 35 post GP)
Boston: 50-52 (68 reg GP, 5 post GP)
Blackhawks: 52-54 (40 reg GP, 7 post GP)
Several played for 5:
Leroy Goldsworthy: NYR, Detroit, Chicago, Montreal, Boston.
Bronco Horvath: NYR, Montreal, Boston, Chicago, Toronto. (was on Detroit roster, played and exhibition game... never played an NHL game for them.)
Bud Poile: Toronto, Chicago, Detroit, NYR, Boston.
Gaye Stewart: Toronto, Chicago, Detroit, NYR, Montreal.
Harry Lumley: Boston, Chicago, Detroit, NYR, Toronto.
Some recent guys with 4:
Mathieu Schneider: Montreal, NYR, Toronto, Detroit.
Anders Eriksson: Detroit, Chicago, NYR, Toronto.
Aaron Downey: Boston, Chicago, Montreal, Detroit.
Robert Lang: Boston, Detroit, Montreal, Chicago.
There are I imagine a lot who have played for 4... especially pre 1967.
Larry Hillman, Terry Sawchuk, Jaques Plante, Lorne Davis, Pete Babando, John Brenneman, Dave Creighton, Reggie Flemming and Carl Voss did, found them after 10 minutes of looking, so there will most likely many more.