What The Eyes Say
Compared to other primates, human muscles are rather weak. In hand to hand combat with a chimp, humans don’t stand a chance. It seems that because we are so good at using tools, we could avoid paying for expensive muscles. Similarly, compared to other animals, language lets humans talk more precisely, and talk about things not in our immediate view. So you might expect we’d get worse at non-language communication. You’d be wrong:
Humans are known to have the largest and most visible sclera – the “whites” of the eyes – of any species. This fact intrigues scientists, because it would seem actually to be a considerable hindrance: imagine, for example, the classic war movie scene where the soldier dresses in camouflage and smears his face with green and brown pigment – but can do nothing about this conspicuously white sclera, beaming bright against the jungle. There must be some reasons humans developed it, despite its obvious costs. In fact, the advantage of visible sclera – so goes the “cooperative eye hypothesis” – is precisely that it enables humans to see clearly, and from a distance, which direction other humans are looking. … Chimpanzees, gorillas, and bonobos – our nearest cousins – follow the direction of each other’s heads, whereas human infants follow the direction of each other’s eyes. (The Most Human Human, p. 39).
Note that most of what we learn via looking at each other’s eyes is hard to verifiably say via language. You might feel he is laughing at you with his eyes, but it will be hard to make that laugh the basis of a group response – others probably didn’t see his eyes at the right moment, or might interpret what they saw differently.
Language was a big innovation, but my homo hypocrites hypothesis is that we humans are now actually post-language in important ways. Language let us express and enforce social norms, but we’ve since developed powerful capacities to coordinate outside the scope of language, to evade those norms. The whites of our eyes seem a key part of that norm-evading capacity.
My suspicion is that Evan is right: the simpler and more likely explanation for the large visible human sclera is that it's just a side effect of having the ability to sweep a large field with the eyes.
The human brain, due to its size, would likely suffer injury if the head were spun as fast as many animals do theirs. Note that the only animals with brains as large as ours have much more massive heads, which probably limit their maximum angular acceleration to safe values.
The human lifestyle demands both high visual acuity, so compensating by evolving a large field of view (like a "fish-eye") is probably not viable. One imagines that the best solution is the one we adopted. The only real downsides I can see off the top of my head (no pun intended) would be a limited maximum pupil size (so poor night vision) and the need for a long, flexible optic nerve.
"By allowing anyone who wants to send signals to aliens, we risk hostile aliens destroying us all. Yet even though the cost to discourage such signals seems trivial . . . ." This is a strange hobby horse of yours. First, many of our activities produce signals as by-products; it would be prohibitively costly to prevent such signals. Second, aliens might interact with us to our mutual benefit. (One crucial question is: what are the probabilities of beneficial and of harmful interaction, respectively? The evidential basis for answering this question is awfully thin. Another question: how likely is it that we will encounter aliens of any kind? I would place this probability so low that the whole issue is practically unimportant.)