Shabbat is not a burden, but a blessing. Gentiles are not allowed to observe Shabbat, can Jews say Shabbat Shalom to gentiles to tell them what they don't have?
There is no problem with the word "Shabat". However, the Gmara in Gitin (page 61) states that one says Shalom to a Gentile "to be in peace with them" ('משום דרכי שלו).
In Brachos (page 17) it is stated that one should always say Shalom to every person, including a Gentile in the market.
So it seems that the answer is a clear "yes".
However, there is a Machloikes between Rashi and Toisfois in Gitin (there). Rashi states the the Shalom that Rav Cahana said was not meant for the Gentile but for his Rav ("שלא היה מתכוון לברך את הגוי, אלא כוונתו היתה לברך את רבו"). And Toisfois says that it's hard to say such a thing since it is Gonev Daas ("צריך עיון שלא יהא בזה גונב דעת הבריות").
Furthermore, there is a machloikes of the Taz and the Bach on the Shulchan Aruch (Orach Chaim, 61), if the word Shalom is considered Kdusha since it is one of Hashem's names. As a consequence, it would be forbidden to utter it the Beis HaMerchatz etc. And so the Bach tends to forbid saying Shalom with no Kdusha intent and the Taz tends to Lehakel, as when his friend's name is Shalom, in which case according to the Taz he can call him even in Beis HaMerchatz. It would seem that their Mechloikes may apply to saying Shabat Shalom to a Gentile. The Mishna Beruru says (there) that it is allowed from Divrai Sofrim, but "Yerei Shamayim" should abstain from saying Shalom in such cases.
Why not? It's just silly because they aren't keeping Shabbat. Would you wish a Merry Xmas to a Jewish person sitting in a Chinese Restaurant on December 25th?