May one say hello to their gentile housekeeper on one of their holidays?
It is permitted to say hello to gentiles on their holidays. Shulchan Aruch writes that one should greet them in a more subdued manner than usual.
See Maseches Kallah, chap. 3, who permits greeting non-Jews on their holidays, without making any stipulation. This is also stated by Rambam (Avodah Zarah 10:5). However, the Gemara (Gittin 62a) writes that upon meeting a non-Jew on his holiday, one should greet him in a subdued manner, and this is the expression used by Shulchan Aruch (Yoreh De’ah 148:9).