June 11.
1184 BC. Troy is captured, sacked and burned by the Greeks. Note that there is considerable disagreement over which century, much less which precise date this occurred. This is the date calculated by Eratosthenes, an ancient Greek. But though he was much closer to the time of the event in question, he was still writing a thousand years after the fact, so take it with a massive grain of salt.
1776. The Contintental Congress appoints a committee of five to draft a declaration regarding its independence. Serving on the committee are Thomas Jefferson, John Adams, Benjamin Franklin, Roger Sherman and Robert Livingston. The selection of the committee and decision of who would try to write the declaration are humorously memorialized in the musical 1776, where Jefferson is portrayed as a young man who ardently desires to return to his wife in Virginia and is instead "forced" to write the declaration by Adams.
http://www.youtube.com/watch?v=vYhjBcYnzvU
1920. At the Republican National Convention in Chicago, party leaders gather in a back room of the Blackstone Hotel to reach a consensus on their candidate for the Presidency, leading the Associated Press to first use the phrase "smoke-filled room". The party's nominee becomes Warren Harding, an unlikely candidate, who goes on to win the Presidency. Harding would have a fairly talent-laden Cabinet, but his administration is marked by corruption and he goes down in history as one of our more mediocre Presidents.
1938. The Nationalist Government of China intentionally floods the Yellow River to slow down or stop Japanese troops. Approximately 500,000 Chinese civilians are killed, as the government decided not to warn anyone about the flooding in order to surprise the Japanese.
1962. Frank Morris, John Anglin and Clarence Anglin successfully escape Alcatraz Prison. Their efforts are memorialized in Clint Eastwood's 1979 movie "Escape from Alcatraz".