Germlish = I can't come to class because I have a holiday.
public holiday = gesetzlicher Feiertag. For example: October 3 is a national/public holiday.
religious holiday = kirchlicher Feiertag. For example: Ascension Day (Christi Himmelfahrt)
to take a day off = sich einen Tag frei nehmen
to have a day off = einen Tag frei haben
What you want to say: I can't come to class because I have a day off.