The proclomation states that his death occurred on June 5. I don't see anyway to argue that he died on June 4...besides, 30 days from that date would carry through July 4, until the colors are retired for that day.
You're right. After reading your post, I recounted (this time taking off my shoes). And since this time I did NOT include a June 31st, I also came up with the 5th. :-)
So flags should indeed still be at half staff on July 4th.