Mr. Hooper's reaction to seeing his own reflection is much the same as everyone else's reaction to seeing him wear the veil.  

[...] his own antipathy to the veil was known to be so great, that he never willingly passed before a mirror, nor stooped to drink at a still fountain, lest, in its peaceful bosom, he should be affrighted by himself.

Mr. Hooper fears to see the veil, just as the members of his congregation do, and though Hawthorne never directly tells us why, we can make some assumptions based on, among other places, the end of the story.

By the story's close, Mr. Hooper is on his deathbed, tended by the Reverend Mr. Clark, who wishes to remove Hooper's black veil now that he is about to pass to the next life.  Mr. Clark says, "'Suffer us to be gladdened by your triumphant aspect as you go to your reward.  Before the veil of eternity be lifted, let me cast aside this black veil from your face!'"  But Mr. Hooper adamantly refuses, and so Mr. Clark desires to know what great sin goes with Mr. Hooper to the hereafter.  Mr. Hooper's final words are the most illuminating:

"Tremble [...] at each other!  Have men avoided me, and women shown no pity, and children screamed and fled, only for my black veil?  What, but the mystery which it obscurely typifies, has made this piece of crape so awful?  When the friend shows his inmost heart to his friend; the lover to his best beloved; when man does not vainly shrink from the eye of his Creator, loathsomely treasuring up the secret of his sin; then deem me a monster, for the symbol beneath which I have lived, and die!  I look around me, and, lo! on every visage a Black Veil!"

In other words, Mr. Hooper has only ever worn a physical symbol of humankind's spiritual state.  Just as the black veil has ever separated him from his fellows, so does the veil each of us holds up between ourselves and everyone else when we do not share our secret sins with each other.  This is why Mr. Hooper tells his community to tremble at each other, not him; they are all sinful in nature -- it is not only he alone who carries the weight of secret sin.  We each try to act as though we are sinless -- though we are all sinners (a popular Hawthorne theme) -- and in that pretending, we lie to one another, preventing ourselves from truly being known by or truly knowing anyone else.  This lack of knowledge leads to a lack of understanding, and so we each perpetuate the myth of our sinlessness, and so others believe that they must do so as well.  No one wants to feel as though they, alone, are a terrible sinner.  And so we deny this part of ourselves, isolating ourselves from our fellows as long as we live.

Therefore, whenever Mr. Hooper would catch his own reflection, he would be reminded of this terrible state of humankind, reminded of his own sinful nature and the sinful natures of all those around him who insist on living a lie.  The sight of the black veil over his own face would only be a reminder that even he was not brave enough to cast aside the figurative veil with which he has, as we all have, separated himself from everyone else.  Thus, the veil is a reminder that he is not only sinful but also, in his own eyes, deceptive and cowardly.

In "The Minister's Black Veil," why does the minister's reflection make him flee into darkness?

Readers know that Hooper can't stand the sight of his own reflection with the veil hanging before his face. He is so averse to himself that he avoids mirrors and even calm, reflective water.

In truth, his own antipathy to the veil was known to be so great, that he never willingly passed before a mirror, nor stooped to drink at a still fountain, lest, in its peaceful bosom, he should be affrighted by himself.

What is not explained to readers and is, therefore, open to reader interpretation is exactly why Hooper is so frightened by his own image. It is possible that he is scared of the physical image of a person wearing a scary black mask. This might make a lot of sense if he was a 7 year old child or he came across someone that he didn't know wearing the veil; however, he is an adult. He knows that he is wearing the veil, and he likely has a pretty good idea of what his image looks like. I think he is averse to seeing his own reflection, because it makes him think of all of the reasons he's wearing the veil in the first place. The veil is symbolic of hidden sin, and when Hooper sees himself, he is viscerally reminded of all of his hidden sins. Running into the darkness is his way of running from those sins as well as eliminating any chance to see his reflection. If there is no light present, then he can't see anything being reflected.

Another interpretation is that the minister's reflection causes him to run into the darkness because, after seeing himself in the veil, he is suddenly reminded of the reason why he is wearing the veil and becomes affected by it. This action answers an earlier musing stated by the physician's wife: “I wonder he is not afraid to be alone with himself.” It appears that he is, indeed, afraid to be alone with himself. He is also afraid of being alone, without a companion. Perhaps the veil reminds him that he has a life of self-imposed loneliness ahead of him. We can see his fear of being alone when he begs his fiancée Elizabeth not to go:

“Have patience with me, Elizabeth!” cried he, passionately. “Do not desert me, though this veil must be between us here on earth. Be mine, and hereafter there shall be no veil over my face, no darkness between our souls! It is but a mortal veil — it is not for eternity! O! you know not how lonely I am, and how frightened, to be alone behind my black veil. Do not leave me in this miserable obscurity forever!”

Once again we learn, this time through words rather than actions, that he is afraid to be alone with himself and that he also wants to have a companion.

The narrator tells us that when Mr. Hooper caught sight of his own reflection, just about to toast the happiness of the newly-married couple, he drops his wine on the ground and runs out into the night because "the black veil involved his own spirit in the horror with which it overwhelmed all others."  In other words, Mr. Hooper had the same reaction to seeing himself as others have when they see him.  When he first began to wear the veil, members of his congregation had a hard time believing that it really was their "good Mr. Hooper" behind it.  Further, one woman said, "'He has changed himself into something awful, only by hiding his face.'"  Children run away from him, he stops receiving dinner invitations, and even his fiancee leaves him as a result of how uncomfortable the black veil makes them.  When Mr. Hooper sees his own reflection, then, he has a similarly visceral response to its horror, and this causes him to run away from the sight of himself.

In "The Minister's Black Veil," what happens to Mr. Hooper when he sees himself?

On the day that Mr. Hooper begins to wear the black veil, he first delivers his sermon, and the topic is "secret sin, and those sad mysteries which we hide from our nearest and dearest, and would fain conceal from our own consciousness, even forgetting that the Omniscient can detect them."  Such a subject allows readers to begin to understand that the veil is a symbol of this attempt to hide our true sinful natures; Mr. Hooper wears it as if to admit that he has such a nature, as we all do. 

After the sermon, Mr. Hooper presides at a funeral, where the solemnity and darkness of his veil seems appropriate, but the wedding he officiates later on is affected in quite a different way.  Here, the bride is overcome by a "deathlike paleness" when she sees her minister, and the groom's hand is "tremulous" when he holds his beloved's.  When Mr. Hooper raises a glass to toast the new couple, he "catch[es] a glimpse of his figure in the looking-glass, [and] the black veil involved his own spirit in the horror with which it overwhelmed all others."  Just as Mr. Hooper's "hearers quaked" when he preached upon the subject of secret sin, the minister himself seems now just as fearful of his own aspect.  Seeing his own reflection, he is reminded of both his own secret sinfulness as well as the secret sinfulness of everyone around him, and, given that (as the narrator said earlier) we all desire to conceal this sinfulness from ourselves and everyone else, the visual reminder of his sinfulness -- in the form of the black veil over his face -- alarms Mr. Hooper enough that he actually drops his wine glass and rushes from the building. 

In "The Minister's Black Veil," what does the minister see in his own reflection?

It would be hard to guess what this eccentric minister sees when he looks in the mirror. It is hard for any of us to see our "real" face in a mirror because we automatically assume an expression that represents our persona--the kind of persona we would like other people to think we are. The same thing happens when we have our photograph taken--unless someone with a camera happens to catch us off guard. It would seem that this minister is wearing a black veil in public because it is the only way he can go about without having to wear a false expression on his face. That expression would probably be about the same as his parishioners would be wearing if they showed their real selves--and they know it. The irony in this story is that the minister is only able to show his true face because he has it concealed behind a veil. The minister cannot even see his real face in a mirror. He is not a likeable character. He seems like an insufferable egomaniac, and we can certainly not blame his fiancee for deciding not to marry him. He must look like a bank robber or a figure in a really bad nightmare. This is Hawthorne's most distasteful story.