I believe that gender is a product of socialization, and therefore you cannot be of a gender that isn't recognized by society.
I get trans women/men, because they essentially are assigned to one out of the two recognized genders (male/female) but identify themselves and want to be socially recognized as the opposite.
Non-binary genders aren't really a part of society. There are no gender roles for a "demi-boy" or a "agender" person. It makes no sense that someone wants to be recognized in one of these categories because these categories essentially don't exist to society.
How is it not just a term for saying that you act outside the gender norms that were assigned to you? How can one identify with these genders if there is no such thing as a social product? And if it is indeed just about behavior/ how is this not offensive towards trans people? How is this not trivializing gender identity??
EDIT: about the "ploy for attention" thing, I'm truly sorry if I offended anyone. As /u/_mach pointed out, this is likely confirmation bias of my part because most of the exposure I've had to non-binary people was on the crazy realms of the internet/tumblr. The fact that some people are doing this for attention shouldn't mean that everyone is.
EDIT2: I think I genuinely have a better understanding about this now. I don't doubt there people exist or that their experiences are valid. I guess I'm still having some trouble with concepts, as I explained in a lot of my comments in this thread. It's not as much as I can't believe a non-binary experience is real/valid, but that I can't really understand how it's more than just a gender performance/expression and actually constitutes a new identity. Most of the comments were super helpful though at broadening my understanding of the topic. Special thanks to all the non-binary and trans people who chimed in on this.
Let me tell you a little bit about my experience. I've always hated activities that divided people up by gender. It seemed arbitrary to me, and I always felt a little out of place in the boys' group. I wondered why people cared so much about it. I didn't really get the point of men's bible study groups, or wanting a confidant of the same gender, or things like that.
It took me something like 20 years of social life to start to realize that other people might feel differently than I do about gender. See, gender isn't really part of my identity at all. I obviously can't do anything but speculate about this, but I suspect that if I woke up tomorrow and had a female body, I wouldn't feel like my self had changed any more than when I change my hair length. I knew this about myself, but I just kinda assumed that it was the same for other people, but they went along with what society expected of them, because that's what you do. It took me that long to realize that some people might actually care about their gender, and might have it actually part of their sense of who they are.
It took me that long because I didn't really have words for these concepts, and because we never really talked about different experiences of gender. So I don't know if "agender" is the right word for it, or if "cis-gender but don't care" is the right word for it, or if "weakly cis-gender" is the right word for it...but we need words to describe different experiences. We need words to be able to tell each other about who we are when it comes to gender in more nuanced ways than "man" and "woman", because those two words do not capture the range of human experience. I have a different kind of sense of self when it comes to gender than some other male non-trans people, and if we don't have language to help us describe that, it's really easy to just assume that other people feel the same way you do, but act differently.