This is a very interesting question. If you are doing a speech based on John Donne's words and you wanted to use Antigone, the Greek tragedy, as an example, then there are several things you can say. Here is one suggestion.
First, you can use the whole play as an example of what happens when people stand alone and do not recognize the importance of others. There is inevitable alienation, death, and great sorrow. For example, Creon does not want to listen to anyone and he decrees that anyone who buries the body of Polynices should be put to death. By refusing to listen to others, he commits a grave crime. This action also forces Antigone to go against the words of Creon and seal her own death as she is determined to bury her brother. She, too, is alone. Finally, the two brother also show isolation, because they fight each other instead of support each other.
From this perspective, the whole Greek tragedy is about people acting like islands to their own demise.