Every now and then one can find some truly useful information on Wikipedia. This list of the world's Independence Days, sent to Tory Historian by a well-wisher, is one of them. It has all the national flags as well, which is an added bonus.
Some countries have several Independence Days. Armenia, for example, celebrated one on September 21 and will be celebrating one on May 28. The first commemorates independence from the Soviet Union in 1991, the second from the Ottoman Empire in 1918.
The Baltic States have separate days to commemorate independence from the Russian (and, possibly, German) Empire and the Soviet Union. Naturally, they do not commemorate liberation from Nazi Germany as that involved enslavement by the Soviet Union.
Tory Historian has been unable to find a country with three Independence Days.