The only one that is not correct (at least by American usage) is "fourth of June." You need to have "the" in front of this phrase. You cannot say, "It is fourth of June." (This may be acceptable in British English, according to another answer, but it seems incredibly awkward and wrong to me.) But you can say, It is the fourth of June. It is June fourth. It is June the fourth. In America, we most commonly use, "the fourth of June" or "June fourth." But you will hear "June the fourth" frequently as well.

In American English we would say June fourth, June the fourth, or the fourth of June. We would not say fourth of June.

In spoken English you can say any of those. British English tends to put the date first and the month second, while American English tends to do the reverse. But basically, there is no such thing as 'more' or 'less' correct. Say whichever feels best for you!