Halacha forbids uttering any Name of God while one is in the restroom, shower, bathhouse or Mikveh, as these are places where people are generally unrobed or perform their personal needs.
The Sages have taught that the word "Shalom," which we generally translate as "peace," is also one of the Names of God, a concept derived from a verse in the Book of Shoftim (6:24 "Va'yikra lo Hashem Shalom"). Hence, one may not say the word "Shalom" in areas where Halacha forbids uttering a Name of God. This applies as well to greetings such as "Shalom Aleichem" or "Shabbat Shalom."
According to the Ben Ish Chai in Parashat VaYetze, Halacha 16, (Rabbi Yosef Chayim of Baghdad, 1833-1909), this prohibition applies even to saying "Shalom" in reference to a person named "Shalom." Since this word is a Name of God, it may not be uttered in these areas in any capacity. Similarly, he rules that one may not mention a person's name if it contains a Name of God, such as "Abdullah" (which contains the word "Allah," the Arabic word for "God") or "Ovadia" (which contains "Ya-h," a Name of God).
Chacham Ovadia Yosef in Halichot Olam, Helek 1, page 43, however, disagrees, and permits making reference to a person named "Shalom" or mentioning a name that contains a Name of God. In his view, since the name is used in this context in specific reference to a person, it may be said in areas where one may not utter a Name of God.
Summary: One may not utter any Name of God, including the word "shalom," in a restroom, shower, bathhouse or Mikveh. This Issur (restriction) would include saying Shabbat Shalom. One may utter the word "shalom" in these areas only in reference to a person known by that name. A name that contains within it a name of God, such as the name "Ovadia," may be uttered in these areas.
