May one say the name Jesus or Yeshu?
From the letter of the law, it is permitted to recite the name Jesus or Yeshu.[1] It is likewise permitted to write these names, as we find Gedolei Yisrael who wrote these names in their Sefarim.[2] Nevertheless, despite the letter of the law, the custom of all Jewry dating back many generations is to avoid saying these names and rather the term Oso Ish‚ or Yoshka, or Yoshke Pandre‚ is used in its stead. One is not Heaven forefend to break this custom.[3]

May one say the word Christ?[4]
One is not to use this term as it connotes a Messiah and savior, and according to some even a deity, and so is the custom of all Jewry to not say this term.[5]

May one say the name Chris-mass?
No, and so is the custom. One is rather to use a epithet [i.e. nickname] such as Kratzmacht; Nittel, and the like. Seemingly, however, the term X-mass is not to be used, as the X is short for Ch***, and is used also by Christians as a formal name of the holiday.[6]

May one say the name Mary?[7]
From the letter of the law, it is permitted to do so[8], although G-d fearing Jews avoid saying this name [when in reference to the mother of Yoshka].

May one say the name Muhammad or other followers of Islam?
Yes, as the religion of Islam is a true monotheistic religion which carries no aspect of idolatry.

_____________________________________________________________________________________

[1] The reason: As it is permitted to recite names of people who do not connote a deity and were later turned into a deity. [Yireim brought in Hagahos Maimanis Avodas Kochavim 5/3; Chavos Yair Teshuvah 1 Hasaga 11-12; Biur Hagra 147/3; Teshuvos Rav Ezriel Hildsheimer 180; See Mishneh Halachos 9/169]

[2] See Biur Hagra ibid; This name is mentioned in various places in the Gemara and Rambam end of Mishneh Torah-uncensored version; See Sanhedrin 43; 67; 105; 107; Avoda Zara 27; Yerushalmi Brachos 5/1; Rambam Melachim However see Teshuvos Rav Ezriel Hildsheimer 180 that no proof can be brought from here that these names may be recited as there is no prohibition to write the names, and rather the prohibition is simply to say them.

[3] Teshuvos Rav Ezriel Hildsheimer 180; See Mishneh Halachos 9/169

[4] Teshuvos Rav Ezriel Hildsheimer 180; See Mishneh Halachos 9/169

[5] Although we find Sefarim that write this term as well, nevertheless one cannot learn from them that it is permitted to say the names, as writing is more lenient than saying, as well as that it is permitted to write the names for learning purposes. [ibid]

[6] See here http://www.thefreedictionary.com/Xmass

[7] Chavos Yair Teshuvah 1 Hasaga 11-12

[8] The reason: In addition to all the reasons of allowance mentioned above regarding Yoshka, this woman is not worshiped or considered a G-d by even the Christians, and hence it has no relation to idolatry. [ibid]