According to "The Minister's Black Veil," yes, all people keep secrets.
Mr. Hooper begins to wear the veil because, as he tells his fiancee, Elizabeth, "'There is an hour to come [...], when all of us shall cast aside our veils.'" The fact that he insists that all of us wear a veil shows that he believes everyone to be hiding the same thing. Further, he tells her, "'If I hide my face for sorrow, there is cause enough [...] and if I cover it for secret sin, what mortal might not do the same?'" So, it is apparent that Mr. Hooper's veil is a tangible symbol of the unseen veil that each of us wears in order to hide our secret sin from others. He wears it to signify that he too, even as a minister, has secret sins that he hides from everyone else, and that this is something all human beings do because we are all inherently sinful (a popular Hawthorne theme).
Mr. Hooper's final words, spoken on this deathbed, confirm this. He says,
"Why do you tremble at me alone? [....] Tremble also at each other! [....] 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!"
Thus, he does not wear the veil to signify that he, alone, has such secret sins on his conscience, sins that he hides from the world so that he can continue to enjoy his reputation as the "good Mr. Hooper." He wears it to signify that we each wear a figurative veil that renders us, ultimately, isolated from all of our fellows because we are never honest with them about our true nature, and they are never honest with us. Mr. Hooper says that he sees such a veil on each face because we are all sinners who try to hide our sinfulness.