For me, I think either phrase would sound okay - but I think it is more common to use the phrase "have a nice day" when you're trying to be polite or talking to a stranger.

They're both equally correct. The choice of adjective depends on the day in question. If it's an ordinary day, you'd say 'Have a nice day'. If it's the other person's birthday, or another special day, you might say 'Have a great day'.

Have a nice day is more common and less enthusiastic. Have a great day is still completely correct but a little more emphatic.

Hellow

In Australia we say "Have a good one!". I don't know if it'll be understood by people from other English-speaking countries though.