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:
ומכל מקום במצוה שצריכין קדושה וטהרה כגון תפילין ס"ת ומזוזה אני חוכך להחמיר שלא יניחו אותם לעשותן
In any case, when it comes to mitzvot that require sanctity and purity, such as [wearing] tefillin or [writing] a sefer Torah or a mezuzah, I deliberated and am stringent¹: we should not allow them [non-Jews] to do so.
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.
I believe that there is a difference between putting on tefilin with or without a bracha. A boy preparing for his bar-mitzvah may put on tefillin without a bracha, since he is not yet obligated to do so, but he may practice putting them on so that he may learn how to do so. In the same way, I believe that a potential convert my try on a pair of tefilin as a learning experience, but he may not make a bracha;neither on the shel yad or the shel rosh.