Answered by Sayyidi Habib Umar bin Hafiz (may Allah protect him and benefit us by him)
Non-Muslims sometimes give us greetings on our Eids. Is it permissible to give them greetings on their holy days?
It is permissible to do so, but not in a manner which implies any meaning related to their disbelief or polytheism, or in a manner that gives the impression of condoning any bad practices that they may perform. We can make the greetings we give them prayers for their guidance. We can, for example, say to them: “May you be given well-being” (`afiyah), which implies well-being in this life and also the next. We can wish them a good day or month (depending on what they are celebrating), intending true goodness, which is that they be guided.
See also this answer on Christmas greetings