May a gentile keep Shabbos?
It is forbidden for a gentile to keep Shabbos. It is forbidden for him to keep Shabbos even on a weekday. A gentile who keeps Shabbos is liable.
May a gentile who is in the process of conversion keep Shabbos?
A gentile who is in the process of conversion may not fully keep Shabbos. He must perform at least one Melacha on Shabbos, to desecrate Shabbos. He may perform even a Rabbinical Melacha. Some Poskim however learn merit on those gentiles who are in the process of conversion who fully keep Shabbos. A gentile who has begun the conversion process, and has been circumcised, but has not yet immersed in a Mikveh, may fully keep Shabbos and is even encouraged to do so.
 Rambam Hilchos Melachim 10/9; Sanhedrin 58b
 Piskeiy Teshuvos 304/2
 Rav Akiva Eiger 121 in name of Panim Yafos Parshas Noach writes that for a gentile the night follows the day, and hence, if the gentile performs Melacha on Motzei Shabbos, it is valid; Binyan Tizyon 91 novelizes that the prohibition is to rest from laborious activity, and hence if the gentile does a laborious activity, even if he does not transgress any Melacha, it is not considered that he rested; Teshuvos Vehanhagos 1/614 writes the prohibition only applies if the gentile thinks that Shabbos was also given to them. If however he agrees that it is only applicable for Jews, then he is not liable.
 Aveni Nezer 351; Binyan Tizyon ibid; Divrei Yosef 2/24; Kovetz Hamiaseif 5665; See Eretz Tzevi 1/41; Shevet Halevi 1/63; Imrei Emes Mishpatim; Michzeh Avarham 54; Imrei Yosher 2/130; Beir Sarim 5/1 and 6/29; Binyan Av 4/59; Minchas Elazar 3/8; Igros Moshe Y.D. 2/149; Tzitz Eliezer 10/25; Lehorors Nasan 1/38; Piskeiy Teshuvos ibid footnote 3