Because that is a song that just about everyone knows, since the movie is set back in the late 1960's. At that time, the Mickey Mouse Club TV show was extremely popular, and just about everyone who had seen the show was able to know the lyrics, as well.
It also points to the innocence of when these soldiers were much younger and there wasn't a war going on. A time in their lives when there wasn't a war and they were all children, opposed to fighting soldiers. Now that they have all grown up, they are singing a song from their youth because it's better than them singing another song which may not have as much innocence in it.
The use of the song Paint it Black is also a reference to the war and how the United States was in a war almost a half a world away. Listen to the lyrics and my pontificating may make more sense.