The phrase "day of June" is correct and usable in written English.
You can use it to describe a particular day in June. For example, "I had the best day on the 15th day of June."
On the last day of June, we had a hundred and thirty-two guests.
May proved to be the worst month for leading shares for three years, and the first trading day of June saw the selling continue.
On the last day of June, 2013, an estimated fourteen million people took to the streets in protest against the government.
As midnight struck on the final day of June, dozens of players spilled into the market, and the Nets and the Knicks began pondering a free-agent bonanza.
**{:.break one} ** We came on board on the 4th day of June and lay the 5th, 6th, 7th, and 8th in the Leith roads getting the ship to a place where we could set sail, which was on the 9th.
Farmington River Tubing in New Hartford, Conn., sent out an e-mail alert the last day of June advertising fast-moving waters and special rates of $15 for people arriving early in the morning or late in the day over the busy Fourth of July weekend.
