Humans need each other; independence isn't about pretending we don't. Independence is having some measure of control over your relationships.* I imagine there are healthier and less healthy ways to go about this. Maybe healthy independence means being able to maintain a standard of how you will interact with others--how you will deal with needing and being needed--and being able to walk away from relationships that insist on violating that standard.
Of course, by definition it also must mean building relationships, of some kind--and keeping them. Because humans, we need each other.
*Credit for this insight goes to Tyrel.