Weekend dates bring in the most people. Others have already suggested June 14th, so I’ll throw in my comment for Saturday, June 28th.

Any plans or ideas for the 27th regarding pride could also be used for the 28th

The Stonewall Riots are said to have started on the 28th since the riots took place in the early morning hours. wikipedia

I’ve seen some groups maintain a chat dedicated to “pop up protests” for these “points of intervention”. They usually only give a week or two worth of notice for these events. They don’t attract the same number of people, but they are still able to get their message across to stakeholders and the press.

For national days of action that we communicate out far in advance, I lean towards dates that let the most people participate.