This is an interesting question. Let me go in reverse order.
The idea of self-reliance is definitely built into the American psyche, because America was built by explorers and people who came to make a new life for themselves. They had to make a new life by working hard with little support. Self-reliance make them survive and strong. The earliest settlers had to work or die. It was a simple as this. When immigrant came, they also did the same thing. This legacy is still with us. Hard work, toughness, refusal to quit are part of America today.
As for Facebook, this is something new. However, what is behind it is very American. This is the idea of the freedom of speech, self-expression, and individuality. One of the great aspects of American culture is its creativity, which is often times rooted in individuality. In this respect, we can say that Facebook represents something American as well.