Captain Picard Day is celebrated on June 16th because it is the equivalent date to Stardate 47457.1, mentioned in the Star Trek: The Next Generation episode entitled The Pegasus. In that episode, Captain Picard Day is an occasion for the Enterprise to host schoolchildren and show them what Starfleet is like.
Celebrate accordingly - it's for the children.
via Captain Picard Day with a tip o'the hat to Daddy Likey. Thanks Winona!
Photo via It's all a dream - the adventures of Jean Luc Picard