We stay at the Shamrock Hotel (Classique Cottage) every Christmas for years. I'm sure they are open at least part of the day on Christmas. I think they may close early or for a few hours Christmas afternoon possibly for the family dinner. We arrived around 6:00 pm last Christmas and the store was open.
The airport on Christmas day last year was the worst I have seen in over 50 trips. The line for immigration was so long you couldn't even see into the room. We have arrived many times on Christmas and don't remember it ever being that bad. I didn't bother having our driver stop for supplies just assuming everything would be closed. We arrived on Christmas Eve a few years ago and the HiLo in MoBay was closing at 3:00 and wouldn't reopen until after Boxing Day. I believe all the grocery stores are closed in downtown Negril but the Chinaman (can't remember the real name) Store near the Burger King should be open. All the bars and restaurants that we frequent are open during the Christmas holidays.