In the United States, the best way to refer to El Dia de Los Reyes Magos is either with a direct translation "The Day of the Three Kings" when referring specifically to the Spanish holiday, or with the corresponding English word, "Epiphany." In Western Christianity, the festivities commemorate principally (but not solely) the visit of the Magi to the Christ child, and thus Jesus' physical manifestation to the Gentiles. Epiphany is celebrated on January 6.
In the United States, we call the day before Christmas (December 24) "Christmas Eve" and December 25 "Christmas Day." We do not celebrate "Boxing Day," and in fact unless you speak to an educated, well-traveled American or an Anglophile, many Americans would not know what "Boxing Day" is.
Growing up, we always received our Christmas presents up to a coupe of weeks before Christmas Day, and the gifts were placed under our Christmas tree. We generally opened our gifts on the morning of Christmas Day (December 25).
PS: Los Reyes Magos are generally referred to as "The Three Wise Men," "The Three Kings," or "the Magi".