Except for some local commuter trains with Green Car, Green Car on Shinkansen and Express trains are reserve seating. Which means, according to the rule, passengers have to obtain a ticket with a train specified and seat assignment prior to boarding a train.

Train conductors may not say any to Gaijin (non-Japanese) because of language barrier (also Japanese tend not to say negative stuff to Gaijin) that JR Green Car Rail Pass passengers may get away with riding on reserved seating cars without obtaining a ticket prior to boarding. But that is against the rule, and Japanese tend to not like people who are not following the rule because Japanese are group oriented. However, same time Japanese tend not to say anything negative right into the face of Gaijin. Although when there is no Gaijin is around Japanese can say something very derogatory to other Japanese, especially if other Japanese are subordinate both in corporate and social sense.

That is my personal opinion, and will stop before my post get into OMNI.