Holidays are days of remembrance of important men or events in history. They are not days set aside for these men or events, though, because to be set aside is to be holy, and holidays are not holy days.
Holy days are days of memorial of saints and angels. They are called holy because they are set aside for the remembrance of those men and spirits whom did the Holy Will of the Lord.
Thanksgiving would be a holiday. It is a day of giving thanks to God, indeed, a day of Eucharistic celebration. But as it is not in memorial of a saint or angel, it is not a holy day.
Christmas would be a holy day, in fact, it is a feast day. You can tell it is a feast because all feast days end in -mas. For example, Michaelmas (Feast of Michael the Archangel) and Martinmas (Feast of Saint Martin). Christmas is the Feast of Christ.
The term Merry Christmas is a salutation of three kinds: First, a warm greeting and a hope for the person greeted to have a merry (joyful) day; second, a simple recognition of what day it is in the year (and possibly a reminder to the person greeted of what day it is); and third, a preaching of the Gospel of the Kingdom of God by reminding the one greeted that Christmas is the day of the birth of Jesus Christ, True God and True Man.
The term Happy Holidays - or Seasons Greeting - is a salutation that attempts to make everyone happy yet without mentioning anyone or anything in particular. Yet, by disregarding those important holy days which many religions observe and by grossly misrepresenting the Season of Christmas (the eight days of Christmastide), the greeter not only offends his listeners but even spreads ignorance, religious indifference, and offense against the Lord. Now not everyone believes in God or follows a religion, yet every man has respect for his fellow brother, and so, that should be a good enough reason to not offend his neighbor. If one wishes to make proper salutations, let him do research on different religions and cultures.