it si possible ona tady nebyla prvního června??
Tady nebyla prvniho červena
Isn't it should be accepted?
It is pretty strange. The meaning is that on several days she "was not" somewhere and the date when she "was not" here (as opposed to other places where she was not on other dates) was the first of June.
Is "v prvním června" an incorrect form?
Yes. It is. The correct form is "prvniho cervna".
What's wrong with: Ona nebyla tady na první červen. Awkward word order maybe?
It's probably wrong because, as I understand it, the genitive case is used with dates.
"na první červen (accusative)" means "for the first of June" as in "there's a celebration planned for that day".