We went this past year - for the last week of May/start of June. The problem wasn't June - it was Memorial Day weekend. Avoid that weekend! I'd suggest getting there the tuesday after that weekend. Also, it was very hot. We had lots of water and portable fans, but it still was uncomfortable. Of course, me being 2.5 months post-partum carrying baby 100% of the time may have not helped! We still managed to have fun though! The gay day and gay days are at the start of June. I did look up the days last time when planning our trip. My thought was that some people might have a problem with it so if they were scheduled for Magic Kingdom, then that might be a good day to go there too! My FIL is not open minded and was there that week with us. We did Sea World the day it was a designated Gay Day. FIL said,"I don't see any gays around,"in surprise. I thought it was rather humorous as if he could tell just by looking at people.
*Edited to add: I forgot to mention, we heard it would likely rain while we were there so we brought ponchos. We didn't need them at all. It only rained twice while we were there - one day that we didn't go to the parks, and one day that we somehow kept managing on being at indoor attractions/restaurants while it rained. The rain never lasted long when it did rain, but it started fast.