FYI, just returned from WDW. Mother and I are DVC members, and usually go this time of year. I am gay, and we did attend the MK on June 1. While there were a LOT of guys there wearing red shirts, I saw nothing inappropriate for a family park, and there were also a large number of traditional families. The park was crowded, but we were able to get on most attractions without too long of a wait, used Fastpass for Splash Mountain and Haunted Mansion. I did see a couple wearing black leather short-shorts. I felt sorry for them in the heat. The guys were much more pleasant and friendly than the crowd the following Tuesday, when Mom and I went back to catch the few rides we missed. On Tuesday people were pushing and shoving, and stopping right in front of her wheelchair, the park was PACKED with people. It was weird, we left and went over to EPCOT for the rest of the day. Mom liked Gay Day better, too.
So, if you don't find gay people totally objectionable to be around, I see no reason to avoid WDW during that time of year, whether you're gay or straight. I'm not trying to make any social statements, just wanted to offer a viewpoint of the parks from someone who was there.