A day before memorial day I overheard 2 girls talking. One girl was saying goodbye and said, "Happy holiday". Is this correct? Is it okay to say "happy holiday" because we're just speaking of one day?
In American English, "Happy Holidays!" is an idiomatic valediction before a holiday/vacation, often said to coworkers and friends:
Note that this is usually only said when both participants are going on holiday, especially before public holidays (esp. major holidays such as Christmas and Thanksgiving).
In British English, "Happy Holidays!" would be understood, but is not idiomatic, and comes across as an Americanism:
British English speakers would more likely say something along the lines of:
Enjoy your holiday!
Have a great holiday!
Hope you have a nice break!
Note that holiday and day are distinct nouns and are only historically related. It is not the case that "holiday" implies a single day vacation and that "holidays" implies a multi-day vacation; you can absolutely "go on a two week holiday to Morocco" and say "Happy Holidays" to an American colleague who is only taking a single day's vacation over Christmas (perhaps because you both work in a hospital and cannot take leave).
So to directly answer your question, I think the most likely thing that's happened is you've overheard elided speech:
[Have a] Happy Holiday!
This is different to the US idiomatic valediction "Happy Holidays!", but has the same meaning; and it would be fully understood by the listener as being a farewell before a vacation begins.
Happy Holidays is used only around Christmas in the USA. Traditionally, it was meant to include both Christmas and New Year's Day. As the years progressed it was used to include other holidays of that season, such as Chanukah, the Winter Solstice and various "Light Festivals" celebrated midwinter by other cultures . We don't usually say "Happy Holiday" at other times, although there's nothing wrong with saying it. We're more likely to say, "Enjoy your holiday" or to say, "Happy _______", and name the specific holiday.
As others have mentioned, "vacation " and "holiday" have different meaning in American English.