Well, "Happy New Year", we say this year-star greeting as a unquestionable habit, which is nice. However, everyday is actually a new start just like the first day of the year.... then, "Happy New Day" can be more used as a greeting instead of "Good Morning" or "Good Day".
On the other hand, this greeting can include the wishes; I wish you a Happy New Year.
Wishing is, generally speaking, acceptable, however, from the energetic perspective, whe you wish, you put an energy also for the reasons of the wish: unhappy past, troubles or so. If so, sort of prayer with wish could trigger more negative results against your wishes. Subtle energy works like this.
And of course, if you do not have any of the reasons, then, it is totally OK, but then, your wishes are not Wishes, anyway: It shall be Expression of Confirmation.
And upon it, it is certain that this new year and coming days are fine to you and us indeed.