'Any one of us' is correct.
In the song lyrics you linked to:
It could happen to anyone of us
Anyone you think of
Anyone can fall
Anyone can hurt someone they love
'Anyone of us' is wrong, but the 'anyones' are right (the lack of punctuation suggests that the writer is not too concerned about the correct use of language).
Enter anyone/any one in the Google Custom Search box near the top of the page for links to more detailed explanations of the difference.