Alright so, I would love to go through different tangents on this, but I'm going to try and give the short version. I would love to go into specifics on different kinds but I'll try to stick with overall themes.
So the first part of the question is do they exist: yes. However, I feel that we need to go deeper than that. Also, I'm going to be using the term Male bias/ Female bias. Here is is meant purely to say that I've seen more art work based around said gender. examples: Slimes generally have a female bias; I've seen much more artwork of female slimes than male slimes.
First we need to define "monster folk", because there are a lot of different categories that could count.
Most people think of Monster Girls fitting into generally the half human half animal all the time (be with either human with minor animal traits, such as with nekos/cat girls, wolf girls, or a literal 50/50 split as with Centaurs, Mermaids etc.). These are generally seen as distinct species and inarguablely fit in all definitions. Many of the original stories about some these have males, and some, like the centaur, originally only showed males. However, there are others whose descriptions (harpies, gorgons) are specifically female. Should also be noted that some of these (Gorgons, Minotaur) are explicitly stated to have started from the wrath of the gods, so only a few individuals were originally afflicted. One version of the Gorgon myth was Athena cursed them for having beauty which rivaled her own. Generally in art these seem to have a female bias.
You also have the more "furry-esque" species such as Kobold, or TES Khajiit or Argonian, and other antropromorphic animals races. This does have overlap with just general furry stuff, but many see them separate. While there are works that treat this concept very seriously, some are done just for fun. A anthro red panda drawn in a detailed fantasy Armour concept art style many would say is fantsy, while the same concept but in a cartoonish style might make other say is "just furry" This also goes into the whole Furry Scale, which is another can of worms. In any case, I've generally seen a more male bias here, but females defiantly do exist. It often times allows for creative difference between genders if they are a sexually dimorphic species.
The most common fantasy races, the non-human bare skinned humanoids who are not part animal, such as orcs, elves, dwarves etc. are sometimes consider monster folk, but often times are seen as a separate category. See Lord of the Rings and most fantasy genera things to be honest. Generally have a more male bias, as the stories they are inspired by generally have this because the society that told them generally had this. That's not to say that all stories that focus on these do have this, and some of them actually have an explanation for this (Often dwarves have legitimate uneven sex ratio at birth and/or both sexes look nearly identical). Dwarves nearly always have a male bias in art (and this extends further than just the males an females look similar) while elves seem to have a female bais in art.
You also have the changed from another group/species. These are things like Werewolves, vampires, and ghosts, angels (sometimes). These range from basically human with minor differences (vampires, some ghosts, angels) to something closer to furries. (Beast form Beauty and the Beast, werewolves). Mostly these have a more male bias. While female vampires do exist, most people think of the "default" as male. Interestingly in Dracula, there are actually more female vampires than male, though these do not have much given personality as vampires, and basically says that Dracula has a Polygynous relationship with them. While I know a few female werewolves exist, most mainstream stuff deals with male ones. Ghosts are generally a mix. Angels are a mix, but can be male or female biased depending on the source.
Finally I'll through in a miscellaneous group. Slimes don't really fit great into the other groups, so they have no problem being called monster girls, but sometimes robots and aliens are seen as fitting here. Due to the overarching nature of this, ans well as some (slimes) clearly being seen as monster girl while others (robots, aliens) not really being seen as such except in things like 30 Day monster girl challenges it's impossible to say which it's more skewed towards. Megaman could be argued is a monster boy series by this logic, which I don't think people realistically consider.
Should also be noted that these groups are not mutually exclusive. Holo from spice and wold is a wolf girl who can transform into a wolf. Some art I've seen of Slime girls are former humans/elves/other fantasy races whose bodies were turned to slime. Vampirism and lycanthropy in theory could afflict races other than bare skinned humanoids like centaurs, or furry-esque being, though I've not seen this concept played with much outside of things like TES, but even then it's usually just the player that has that, not npcs. I've also seen art of Cat/dog style centaurs, where the horse part is the carnivore, but the head and torso are of an anthropomorphic version of the same animal.
So anyways, as for the main question of "are monster boys a thing": There are a lot of males in the Bare-skinned humanoids and the changed groups. The Twilight saga is technically a a monster boy franchise, and the same could be argued for Lord of the Rings [and very debatly Megaman as I said above]. However, I'm 100% certain this is not the answer you are looking for
I seem to get the implication with "are people, specifically girls, okay with the idea of monster boys or men?" that the question is more aimed at things like "is there a male version of something like Daily Lives with Monster Girls" where the purpose is more fanservicy. The Twilight saga could be argued [I'm not trying to argue quality just that non-human and humans having romantic relationships are the focus unlike say Lord of the Rings], but again that's probably not what you mean.
As for other stuff and "do girls like it", I know that Lisa Lombardo (ikr) was upset that most monster boy stuff was just boring vampires and other technically not humans and wanted more creative stuff like Monster Musume had. Also, just by change I stumbled across this female DA Artist who drew male Lamia (see the side of the image page for more of them). Some of the situations do have homosexual implications (though there's also a lot of that in monster girl works too so take that for what you will) from OP "because monster boys are gay."
Really though, people are different and have different likes and dislikes. Despite the massive post I made, I don't have the same interest in Monster folk as most people who are big fans of Daily Lives with Monster Girls.