We usually say that "좋은 하루이길 바래" or "좋은 하루였으면 좋겠어" when they wish someone a good day. And there is technically no difference between 좋은 날 and 좋은 하루 for meaning of "a good day" in Korean. Korean use both 좋은날 and 좋은하루 in the situation. I tend to prefer to use 좋은 하루.

Maybe you know that specific things have different counters in Korean. For example, things in general have 개, animals have 마리, people 명, bottles 병 etc. So do days. 하루 is the day counter (it means one day). It goes as: 이틀 = Two days; 사흘 = Three days So 하루 is only used when you count days. (Mostly people simplify it by using 일 though, although 일 is just supposed to be an exact date, like 28일 - the 28th of a month) 날 means a special day of something happening, like when you refer to one exact special day. So you mostly wish 좋은 하루, unless it's like the special day of birthday or wedding which is exclusive to that one day, then it's 날. If you hope for something to happen in Korean, you add -면 좋겠어. So: 좋은 하루 있으면 좋겠어. Literally: Good_day_is + if_would be good.