I'm not sure the premise of the question here is accurate for two reasons.

1. Veiling suggests a covering of the face. There is - to my knowledge anyway - no religion that mandates a full face covering. Even the examples you cite are not veils covering the face, but various forms of hair coverings (so hijab, "Christian veils", habits, etc.). While the niqab and the burqa both cover the face in some Islamic traditions, they are not a majority by any means, even among Muslims. But I'm going to assume that this is just an issue with terminology, which brings me to my second point:

2. There are several religions in which men are instructed to cover their heads for religious purposes. Two of the most famous would be Judaism (Yarmulke a.k.a. kippah) and Sikhism (Dastar a.ka. Turban). In both cases the wearing of a head covering is mandated by the religion itself for various reasons, but mostly as a sign of devotion to God. In fact, the term yarmulke even means "reverence to God" in Hebrew Yiddish, and the requirement is not limited to Jews alone. If you've ever been inside a synagogue - especially a conservative or orthodox one - you were very likely given a paper or simple cloth kippah to cover your head while inside the grounds/building. As for Sikhs, I know less about that religion, but as I understand it, baptized Sikhs are forbidden from cutting their hair, again as a sign of devotion to God, and are required to wear the Dastar as a sign of piety and devotion. I should note, as well, that among Catholic and some Orthodox clergy (so among men), a form of skullcap called a Zucchetto is also quite common, although whether its mandated or not, I couldn't say.

edit: words
edit 2: thanks for the correction u/alice-in-canada-land