Disney released the June 2013 operating schedule here. As usual, click refresh if the date at the top doesn’t say “June 2013.” Also as usual, these are the official operating hours that likely won’t be available at DisneyWorld.com until closer to the end of the month.
I’d look for Star Wars Weekends to be scheduled on May 31 – June 2 and June 7 – 9. Otherwise, June is a month that gets progressively busier and hotter as it gets later in the month. On the plus side, there’s the addition of a second set of evening Extra Magic Hours at Magic Kingdom and the operating hours are among the longest of the year. Entertainment is also scheduled more often than any month other than July.