These 3 days in June were very significant in World War II. To start with, June 4, 1940 was the last day of the Dunkirk evacuation, the most famous such sea-borne evacuation in history, though only the third largest of the war. The British withdrew even more people in the flight from the rest of France in June 1940 as the country collapsed. And both were dwarfed by the German flight from the east in 1945, in which approximately 1.5 million were evacuated, with a loss of perhaps 2%. It was probably the Kriegsmarine's finest hour.
June 4, 1942 was also a very momentous day -- the day of the major carrier battle in the Midway campaign. When the day was done, 4 Japanese carriers had been fatally stricken, as well as the USS Yorktown. Japan never again had naval superiority in the Pacific, though it would win many more naval actions over the next year or so.
And on June 4, 1944, the American Fifth Army under Mark Clark occupied Rome. It was their big moment of publicity (which Clark, like so many higher commanders, craved) -- though only for a brief moment.
The reason their moment didn't last long is the great Normandy invasion. This was actually scheduled for June 5, 1944, but bad weather caused a one-day delay. The Allies landed on 5 beaches, and successfully made a lodgement on each. Within another day, all the beaches except the westernmost (Utah beach) would be linked up even as the first German reinforcements (the !2 SS Panzer Divsion) arrived. It would take nearly 2 months for the Allies to break out by way of the VII Corps' Operation Cobra.