Yes, I agree with the saying that "No man is an island", as what this saying means is that every person affects the others around him/her. To suggest that someone is an "island" would mean that this person does not exist in the greater world, and that is patently untrue.
It has been proven that people psychologically depend on one another. That is why solitary confinement is such a terrible punishment, and one that drives people insane. People require human contact, even if they don't require very much. We are programmed to want to be around each other, and help each other, and interact with each other, and so yes, I agree that no man is an island.