Czwarty lipca! :)
Doesn't this have to be piątego czerwca?
Not without a context. Here we just have a noun phrase, there's no even a dot to indicate that it's some short answer to a question, so we use Nominative.
Czerwca is not one of the possible answers for "of June" in the questions?
If you mean the hints, I just added it. If you mean that there was no tile for "czerwca", that seems pretty unlikely, it would be a terrible bug.