I know that 私, while being used by both men and women, is relatively formal or at least polite and so wouldn't necessarily be used when talking to a close friend, etc. On the other hand, 俺、僕、and あたし are all gendered pronouns. Is there a first-person pronoun that is gender neutral but still casual that someone who is non-binary or otherwise not wanting to express their gender could use?
As @user4092 stated in the comment above, there exists no such first-person pronoun in the so-called Standard Japanese. If there existed one, someone would have answered this question as soon as you posted it.
I could think of two such pronouns used in other dialects. One of them is 「わ」 used in Tsugaru dialect (Aomori Prefecture). This dialect is known for its numerous extremely short words as 「わ」 might already suggest.
The other is 「わし」 used widely in the western half of Japan, but I must also mention that it is not used by "everyone" in Western Japan at least the way "I/me" is used by English-speakers. The socioeconomic and other factors may well prevent people from using it. To be also noted is that 「わし」 is generally used by older people.
The short answer, as others have pointed out, is 'no'.
But people switch first-person pronouns depending on context all the time. In a formal situation I'll use わたし, and otherwise usually おれ or maybe ぼく if I'm talking to small children, for example.
Not speaking from personal experience here, but if you (or the person you're asking for) don't identify as unambiguously male/female then I think you have three options.
1) Use the more formal わたし in casual situations.
A lot of non-native Japanese speakers do this anyway, so it won't come across as too strange. Where you stand on the formal/casual continuum depends on a lot more than just choice of first-person pronoun, so you can fine tune that with body language and other linguistic choices (to です or not to です, for example).
2) Use a pronoun that clashes with the gender people might otherwise automatically assign you to.
That is, use おれ or ぼく if people are likely to assume that you are female, or あたし in the converse case. The former is actually reasonably common (although not the norm), at least when close female friends talk among themselves, while the latter is more likely to raise eyebrows.
3) Use your name as a kind of pronoun.
Say things like ジョンもカラオケに行｛い｝きたいなぁ。Instead of saying (the quite odd) わたしの名前｛なまえ｝はジョンです just say ジョンです。You get the picture.
Option 1) is probably the better option for keeping things ambiguous, might work better in what I think of as 'faux casual' situations, where there is no formal hierarchy but people aren't really soulmates yet. On the other hand option 2) challenges assumptions a bit and may be best reserved for people who know and understand you, unless you like challenging assumptions (or happen to feel like it in a given situation). If you really want to keep people on their toes you could switch randomly, although this will probably come across as a bit odd. Option 3) sounds quite childish to me. My four year old speaks like this, but my seven year old has switched to ぼく. A lot of adults (particularly women?) do it (sometimes), but it strikes me as quite "cute" which is probably the intention in most cases, but may not be what you are trying to convey. On the other hand, if you do it enough your friends will probably get used to it and sometimes it doesn't hurt to adopt a few childish mannerisms when you are learning because people are more likely to speak to you using clear, simple language. Also, "cute" is a package -- if the rest your presentation is not "cute" then the net result won't be "cute".
What I'm trying to emphasize is that this is a question of choice, rather than trying to find the 'right' answer. Unfortunately a prepackaged choice with the nuances you would like is not available, so you'll have to work with the resources that the language provides. Feel free to experiment and make "mistakes" - it is part of the learning process. But as you do, bear in mind that these choices are not just a statement about you, they also reflect your relationship with the person that you are speaking with or - more accurately - your interpretation of that relationship. So, as with so many other things in Japanese, context is everything and you might make different choices in different situations.
Also note that according to careful observational studies, students of Japanese tend to use first person pronouns about 3247 times more than is actually necessary. Nothing says 'ambiguous' like not saying it at all and there are very, very few situations where the first person pronoun is actually necessary.
Finally, if you can, seek out other people in a similar situation -- on TV or YouTube if you can't find them in real life -- and see how they present themselves in different situations.