Aegisthus is the current King of Argos, having murdered Agamemnon fifteen years earlier to usurp the throne (and the Queen). Like every other character in The Flies, Aegisthus acted in accordance with his own value system. Orestes values freedom; Zeus values remorse; Electra values revenge; and Aegisthus values order.
[A passion] for order? That is so. It was for the sake of order that I wooed Clytemnestra, for order that I killed my King; I wished that order should prevail, and that it should prevail through me. I have lived without love, without hope, even without lust. But I have kept order. Yes, I have kept good order in my kingdom. That has been my ruling passion. (2.2.103)
And, like every other character in the play, Aegisthus isn't morally condemned for his particular value system. There's nothing wrong with a desire to maintain order. What's problematic for the existentialist, however, is that that order comes at the expense of personal freedom:
Each keeps order; you in Argos and I in heaven and on earth – and you and I harbor the same dark secret in our hearts. […] The bane of gods and kings. The bitterness of knowing men are free. Yes, Aegisthus, they are free. But your subjects do not know it, and you do.
Why, yes. If they knew it, they'd send my palace up in flames. For fifteen years I've been playing a part to mask their power from them. (2.2.92-5)
By imposing her value system on the Argives, Aegisthus never allows them to choose their own system. In fact, he hides from them the possibility of choosing for themselves. In this way, he strips them of their freedom.
Sartre makes a point of explicitly connecting Aegisthus and Zeus time and time again throughout The Flies. As we've seen, both represent institutions of order that force their value systems onto others at the expense of freedom. Both maintain power by hiding freedom from their subjects. And both suffer from a rupture between their actual selves and the image they project before those they rule:
For fifteen years I have been dressing a part, playing the scaremonger, and the black of my robes has seeped through to my soul. (2.2.33)
What am I but an empty shell? […] Looking within, I see I am more dead than Agamemnon. (2.2.41)
Since I came to the throne, all I said, all my acts have been aimed at building up an image of myself. I wish each of my subjects to keep that image in his mind. […] But I have been trapped in my own net. I have come to see myself only as they see me. […] Almighty Zeus, who am I? Am I anything more than the dread others have of me? (2.2.97)
Good question. One of the issues Sartre explores in Being and Nothingness is that of the roles we play. If you try to answer the question, "Who are you?", you might say that you're a student, a sister, a flute-player, an American, or a baseball player. Each of these is a role you play in the world, but none of them is who you are. Sartre believed that our existence was something beyond these labels. In other words, he claims that there is a you who exists outside of the roles you play.
By now you've probably heard us talk about Sartre's different kinds of being. One is being-for-itself – the type of conscious, active being that humans have. When you say "Tom is," what you mean by the verb "is" is being-for-itself. Another type of being is being-in-itself – the type of unconscious, passive being that applies to objects. When you say, "The apple is," what you mean by the verb "is" is being-in-itself.
Now take a sentence like "Tom is a baseball player." For the existentialist, this is a problematic statement. To which type of being are you referring with the verb "is"? Because the subject of a sentence is a person, you should be indicating being-for-itself (the conscious, active type of being). But Tom can't be his role (that of a baseball player), because Tom is something that exists beyond the various roles he plays. Tom simply is.
Now what the heck does this have to do with Aegisthus? It's fairly clear that the role we're talking about here is that of a king. Just as "Tom is a baseball player" is a problematic statement, so is the sentence "Aegisthus is a king." It's particularly problematic for Aegisthus since he has worked so hard to cultivate the image of a king – to the point where he has no sense of self outside of his role. No wonder he's miserable.