The saheeh Sunnah indicates that it is not permissible to initiate the greeting with the Jews and Christians, as Muslim narrated in his Saheeh (2167) from Abu Hurayrah (may Allah be pleased with him) that the Messenger of Allah (blessing and peace of Allah be upon him) said: Do not initiate the greeting with the Jews or Christians.
The Sunnah indicates that we should respond to them when they greet us, as al-Bukhaari (6257) and Muslim (2164) narrated from Abdullah ibn Umar (may Allah be pleased with him) that the Messenger of Allah (blessings and peace of Allah be upon him) said: When the Jews greet you, one of them says al-saam alaykum (death be upon you).' So say: wa alayk (and also upon you). [NB: The Jews of Madinah played with the words to insult the Muslims]
The same may also be said about shaking hands: we should not initiate shaking hands with a kaafir, but if he holds his hand out, we shake his hand. Shaykh Ibn Uthaymeen (may Allah have mercy on him) said: Ibn al-Qayyim (may Allah have mercy on him) said: If this reason no longer applies, and the kitaabi (i.e., Jew or Christian) says, Peace be upon you and the mercy of Allah [i.e., it is a sincere greeting], then fairness dictates that we should respond with a similar greeting.
End quote from Ahkaam Ahl adh-Dhimmah, 1/200
In Saheeh al-Bukhaari it is narrated from Ibn Umar (may Allah be pleased with him) that the Prophet (blessings and peace of Allah be upon him) said: When the Jews greet you, they say, as-saam alaykum (death be upon you),' so say, And also upon you.' If he holds his hand out to you, then reach out and shake his hand, otherwise do not initiate it. End quote.
And Allah knows best.
Reproduced from Islam QA