If I had a sterilized dead cockroach, would you be willing to look at it? Pick it up? Touch it to your lips?  Drink a glass of juice after dipping the cockroach in it? 
Psychologists have found that we tend to be less than enthusiastic about these actions. The reaction of disgust we have to an object, like a cockroach, tends to spread to other objects that it comes in contact with. Disgust, in the parlance of psychologists, acts like a contagion. In some cases, subjects can avoid the disgust reaction by taking some positive step, like washing their hands. In other cases, subjects remain disgusted by the object.  This second kind of contagion, in which the disgust reaction spreads and cannot be eliminated, is often referred to as "magical thinking." We know the cockroach can't contaminate the juice since it's been sterilized, but still the juice revolts us.
In many cases, this kind of "magical thinking" is irrational and harmful. But we shouldn't be so quick to dismiss this phenomenon. While disgust can be massively destructive, serving as an impetus to morally abhorrent ideologies and actions (racism, anti-Semitism, homophobia, etc.), might it also serve a constructive purpose in moral life?
The cockroach example seems silly, but consider some other cases closer to the heart of morality. Suppose you are researching Nazi Germany and find some photos in an archive of a lamp. As you inspect the photos, you might have a passing curiosity, wondering who owned the lamp and why it was photographed. On closer inspection, you come to the realization that the lamp is in fact made with human skin.
What is the morally appropriate reaction to coming across such shocking evidence of moral depravity? The answer is no doubt complex. One might appropriately feel rage at those responsible for these crimes against humanity, sadness at the depth of evil that has been perpetrated, compassion and solidarity for the victims robbed of their lives and dignity, and a passionate commitment to do one's part to prevent future atrocities. However, there's another reaction that seems appropriate, perhaps even demanded: a visceral, stomach-turning repulsion that presses one to look away from the photos, put them down, and move away from the archive. Not only does this seem perfectly rational, it may be a moral failure not to have some such reaction.
A claim of that sort might seem out of place in ethical theory. Are there really moral principles governing when we ought and ought not feel disgusted? In the context of an ethics of virtue and character, however, the claim is well motivated. The disgust reaction is a powerful psychological force leading us to avoid certain actions and objects. In short, it is a powerful tool for developing a morally sensitive character. It is also more than that: it is a failure of one's moral sensibilities not to be repulsed by certain kinds of heinous crimes. Someone who does not have those emotional and physiological responses is out of touch with moral reality, just as someone who is hallucinating is out of touch with the reality of the external world. Whether in a virtue ethics or a sentimentalist ethics, disgust can have a key role to play.
Not only does the disgust reaction have an important place in the moral life-- albeit a precarious one, given the destructive role that disgust can play-- but it also has an important counterpart in our moral discourse. The everyday vocabulary of moral discourse, unlike the carefully selected bits of that discourse that tend to be focused on in moral philosophy, is incredibly rich with terms of both appraisal and condemnation. Certain kinds of actions are not just bad, wrong, or unjust, but vicious, depraved, wicked, or disgusting. Philosophers have paid more attention to such thick moral concepts in recent years, but "disgusting" as a term of moral condemnation is a particular kind of thick term. Not only does it mix the descriptive and evaluative, but it does so by reference to the reaction of disgust itself. An action labelled disgusting is one that, according to the speaker, one should find disgusting. (The much weaker reading, on which a speaker's saying "X is disgusting" only entails that the speaker is disgusted by X, doesn't do justice to the case of moral disgust we considered earlier.) This normative claim about how one should react to certain kinds of actions is an additional normative element to thick terms like "disgusting," since to call an act "disgusting" is also to find a more general fault in it (it's bad, or vicious, or unjust, for example).
Unpacking and justifying claims about what one should find morally disgusting is a difficult task, just as it is hard to specify when one should be angered, or saddened, or regretful. Disgust, no less than these emotions, can be felt appropriately or inappropriately, and has a crucial role to play in our moral lives, even if we can imagine beings for whom some of these responses do not play such a role. We should ask the same question that Hume put to Hutcheson:
"If morality were determined by reason, that is the same to all rational beings; but nothing but experience can assure us that the sentiments are the same. What experience have we with regard to superior beings? How can we ascribe to them any sentiments at all? They have implanted in us for the conduct of life like our bodily sensations, which they possess not themselves." 
 Rozin, Haidt, McCauley. "Individual Differences in Disgust Sensitivity: Comparisons and Evaluations of Paper-and-Pencil versus Behavioral Measures." Journal of Research in Personality 33, pp. 330-351(1999).
 Rozin, P. "Technological Stigma: Some Perspectives from the Study of Contagion." In J. Flynn, P. Slovic and H. Kunreuther (Eds.), Risk, Media and Stigma – Understanding Public Challenges to Modern Science and Technology, (London: Earthscan Publications Ltd., 2001). pp. 31-40.
 Nemeroff, C., & Rozin, P. (1994). The contagion concept in adult thinking in the United States: Transmission of germs and interpersonal influence. Ethos: The Journal of Psychological Anthropology, 22, 158-186.
 Letter dated March 16, 1740. Life and Correspondence of David Hume, ed. John Hill Burton, Volume I (Edinburgh: William Tait, 1846), p. 120.