May a non-Jew or a person who is in the conversion process don tefillin? If not, what are the reasons that they cannot?
A non-Jew certainly may wear tefillin (in other words, there is no law against them doing so), but they will not be fulfilling a mitzvah. From that perspective, they might be viewed in the same way that one views a Jewish woman who lays tefillin: the Shulchan Arukh (Orach Chaim 38:3) exempts her but allows her to wear them if she wishes. Note, however, that the Rema does not permit women to wear tefillin (although he does not comment on a non-Jewish man doing so), and you can see the explanation of the Magen Avraham for this in situ.
The Shulchan Arukh prohibits non-Jews from writing the scrolls that go into tefillin (Orach Chaim 39:1), but it also mentions that if a non-Jew is in possession of tefillin one makes the assumption that they are kosher. Nowhere, however, does it refer to non-Jews who choose to wear them.
That said, the Rambam does speak about non-Jews choosing to perform mitzvot in which they've not been commanded (Hilkhot Melachim 10:10), and even suggests that they receive some measure of reward for doing so. If you look at what the Radbaz has to say on this point, you'll see that he mentions tefillin in particular:
In other words, while the law appears to be in favour of them doing so, if they should wish to do so, and while it may even be possible to construe the Rambam's words as suggesting that they even receive some degree of merit (despite not performing a mitzvah), there is also scope for a less lenient position, as per the Radbaz, and as per a possible extrapolation from the writings of the Rema and the Magen Avraham.
¹ The expression that I have translated above as "I deliberated and am stringent" appears in the Mishna, Nedarim 1:1. It literally translates to "I hesitate to be stringent", but its meaning is as I have rendered it above (cf: Jastrow, חכך II; Kehati, Ned 1:1).
Rabbi Barry Fruendel, who has been representing the RCA in negotiations with the Israeli rabbinate to establish standards for American conversions acceptable to the Israel rabbinate, instructed one convert I know to put on tefillin and to wear a tallis (he was married) before completing his conversion. Frankly, I was suprised. When I was working to get an Orthodox conversion -- I had a Conservative conversion first and thereafter wore tefillin -- I had to stop wearing tefillin until I completed my conversion.
Tefillin as I understand is a mitzvah which teathers your body to the 'light'. That is, you literally put them on your arm, which represents the heart, and on your head, which represents the eyes. In doing so, and saying the customary blessings and prayers with them enough, you gradually fill up your eyes and heart, or your organs, with what they call 'light'. It's the same with any mitzvah - each mitzvah has a specific vessel where the mitzvah resides. Once you begin to fill up that place of residence, you begin to transform what that place is.
For instance, there's my 'body', and there's how I perceive my body or how I approach my body. There's how I surround my body. If you'll notice when you're wearing clothes, you're not really that conscious of your nakedness, even though your body is the same whether you are wearing clothes or not.
So what gives? Basically, your mind produces simplification techniques so that it doesn't need to constantly be processing everyone and everything in your environment. To do so would be overwhelming and you could never make decisions. So it cheats, and it comes up with forms, and layouts, and instead of picking up on every single detail, it starts to abstract the details, to unify them, and to aggregate them. For instance, right now I am wearing a "shirt". This is an awfully short word for something which is comprised of thousands of pieces of fiber, each which has its own characteristic. But because it is worn in such a way, and looks a certain way, those fibers have a certain pattern, and that pattern in English is called a 'shirt'. Then, I don't need to know all the details which make up the shirt - I can just accept it as a shirt and love on.
Similarly with the mitzvah of tefillin. The mitzvah calls for a person who aspires to achieve a certain "form" in their life, that they need to and are commanded to bind the message of the unity of the creator to both their eyes and and their heart - so that when their eyes see, they will only see the unity of the creator- when their heart beats, their heart will only beat the blood of unity. And so on. Does a person need to know the details of everything that they see? It's color, wavelength, hue, placement, shading, pattern, texture, etc? Those things are important, but what tefillin is trying to bring out is that everything that the person sees with their eyes is actually the creator, just in a very non aware that it is the creator.
So to the question - should a non Jew or a person in the process of converting put on tefillin - I think the person should understand and know why he/she is converting, and what it means to convert. They should understand that by donning tefillin, they are asking to perceive the chaos of their immediate world in such a way, and are asking for it to be organized in such a way. It glues the matter that they are able to find, perceive in such a way, and the patterns that they are able to identify. That this will significantly effect who they are and how they understand their self / how they are able to find or define their self.
If the person is willing to do these things, I say they should go for it. Goodness knows there is enough practice done on the basis of habit or simple obedience that lack an understanding of mitzvot, why the mitzvot are the way they are, what is a mitzvah, how can they be practically understood, and what is the point of them in their entirety or in general.