It's hard to ignore what seems to be an obsession with eyes in this play. Eyes take on several different meanings and get at several different points in Sartre's philosophy. We'll tackle the big issues one at a time.
First of all, eyes are part of physical appearance. As we discuss in "Tools of Characterization," external appearance is a great indication of internal changes throughout the course of The Flies. Let's start with Orestes. Electra is drawn to his youthful innocence and boyish good looks in Act I. In Act II, she says to her brother, "You came here with your kind, girlish face and your eager eyes – and you made me forget my hatred," insisting, "Yes, it was your eyes that made a fool out of me" (2.1.115, 107). In her mind, eyes an indication of character. If this Orestes has such gentle and kind eyes, surely he can't be the vengeful and angry Orestes of her imagination. When he claims that, actually, he can be the vengeful and angry Orestes, she is confused. "To think it's you who are going to shed [their blood]," she says, "you with those gentle eyes!" (2.1.180)
Now look at what happens after Orestes murders the royal couple:
Oh, how you've changed! Your eyes have lost their glow; they're dull and smoldering. […] You were so gentle. (2.1.170)
There are two ways to interpret this. On the one hand, it could be that Orestes's physical appearance has actually changed in the course of a few hours. In this sense, the eyes really are the window to the soul. As he has altered completely his fundamental project, so that change manifests itself physically on his face. Another possibility is that Electra projects her own thoughts onto Orestes. She now imagines this man standing before her as the same vengeful Orestes of her dreams, and so she sees him physically conform to her standards.
Moving on to our next point, let's talk about the repeated references to "dead eyes" throughout The Flies. It starts when Orestes first sees his mother: "I hadn't counted on those dead eyes," he says (1.1.171). Clytemnestra then notes the "smoldering eyes" of her daughter and laments that she, too, used to look that way (1.1.180). By Act III, Electra's "bold eyes" have transformed completely. As Orestes gazes upon her, he says, "Where, now, have I seen dead eyes like those? Electra – you are like her. Like Clytemnestra" (3.1.32). Electra, like her mother before her, has lost the passion and the verve of youth. Orestes, too, has moved past his youth – but he has moved towards a committed and authentic sense of completed self. Electra, on the other hand, is just "dead." Because she can't fully commit to the value system she has chosen, and because she is fleeing from personal freedom, she isn't fully alive.
Next is the idea of other people's eyes and the fear of judgment. If you like what you've seen in The Flies, you might check out Sartre's other famous play, No Exit, which is about the idea of "the other" and "the look." Sartre argues that the gaze of other people is a horrifying experience. When other people look at you, they turn you into an object. You lose your subjectivity as you realize that this other person is his own subject, and that in his eyes you are merely a thing. This idea of "competitive subjectivity" is the focus of No Exit.
The Flies, however, focuses more on the idea of judgment. When characters like Electra or even the general Argive population choose to flee from their freedom, they give their freedom up to others. They allow others to view them, judge them, and create their identities. When Aegisthus promotes the myth of the Ceremony of the Dead, he is playing off of this idea:
Well you may cry for mercy! Play your parts, you wretched murmerers, for today you have a full house to watch you. Millions of staring, hopeless eyes are brooding darkly on your faces and your gestures. They can see us, read our hearts, and we are naked in the presence of the dead. Ah, that makes you squirm; it burns and sears you, that stern, calm gaze unchanging as the gaze of eyes remembered. (2.1.52)
In this way, eyes come to represent judgment symbolically. Even Aegisthus falls victim to his own myth when he asks Clytemnestra, "Are you not ashamed – under [Agamemnon's] eyes?" (2.2.37)
Electra recognizes eyes as judgment after Aegisthus is dead. "No," she says, "I can't bear those eyes any longer" before placing a mantle over the dead king's face (2.2.136). Similarly, Electra tries to exert judgmental control over Orestes: "I must see you," she says; "when I stop seeing you, I'm afraid of you. I daren't take my eyes off you" (2.2.145). And of course, the furies use their gaze as a threat against would-be-penitents: "We shall be the staring eyes of the houses," they say (3.1.16).
We've talked about eyes as a reflection of internal changes, as indicative of humanity (or lack thereof), and as vessels of judgment. But we can't forget the most obvious meaning of eyes in The Flies – they are how we see. When eyes are covered or closed, it means that the viewer isn't fully aware of his surroundings. Orestes's Act II transformation, then, is akin to opening his eyes to the anguish and the burden of human freedom. When he seeks to free the Argives, he seeks to open their eyes.
The folks of Argos are my folks. I must open their eyes.
Poor people! […] You will tear from their eyes the veils I laid on them, and they will see their lives as they are, foul and futile, a barren boon. (3.1.137-8)
And that just about covers it.