1st day of month you win the game if you are the first person to say rabbits
The informant said that this is a game within his family and he thinks its from his father, who is from England, so that it might be English.
Im not certain on the significance of the game, but I know that rabbit is a much more popular food overseas in England, than it is here in America. England also has a booming rabbit population and rabbit hunting is quite popular. So, perhaps, this game represents a rabbit sighting, which would lead to a rabbit dinner. In saying rabbits! one may expect to see rabbits, which could then be hunted. Therefore they win because the sighting would lead to a tasty dinner for the family.