Allah tells us in Quran to spread the "salams" (greetings of peace) by always replying with something better than what is given to us, or at least as much.
The explanation about this comes from a teaching (hadeeth) of mankind's last prophet, Muhammad, peace be upon him), demonstrating to us how to reply to the greetings of peace as follows:
* Someone says, "Peace be to you" (Assalam alaykum); you say, "And unto you be peace and the Mercy of Allah" (Wa alaykum salam wa Rahmatullah".
* Someone says, "Peace be to you and the Mercy of Allah" (Wa alaykum salam wa Rahmatullah); you say, "And unto you be peace and the Mercy of Allah and also His Blessings" (Wa alaykum salam wa Rahmatullah wa Barakatah).
* Someone says, "Peace be to you and the Mercy of Allah and also His Blessings" (Wa alaykum salam wa Rahmatullah wa Barakatah); you say the equivalent back to them and that is enough (Wa alaykum salam wa Rahmatullah wa Barakatah), and that is sufficient.
It is also recorded in the most famous and classic of all hadeeths because of these being actual written letters dictated to the rulers of the empires of Rome, Persia and others, the Prophet of Allah, peace be upon him, dictated to his scribes to incorporate the greetings of peace to all who seek guidance of truth. The people were definitely not Muslims, because the statements in the letters asked them to accept Islam for success in this world and the Hereafter.
This clearly indicates it is correct to give at least the minimum greeting of "Peace be unto you" and to reply to the greetings with the minimum of "And unto you be peace".
There is a note here in connection of what to say to one who sneezes but is not a Muslim, "Yahdikum Allah wa yusli balikum" (May Allah Guide you and better your affairs); indicating a very lovely supplication made on their behalf to Allah to be rightly guided and be in a better condition and circumstance.
Some of our brothers and sisters are confused as to what to say to non-Muslims when they offer our greetings of "Peace be unto you" (Assalamu alaykum); due to misunderstanding another hadeeth (teaching), telling about one occassion when some of the disbelievers were making fun of him by greeting him with something that almost sounded like the Arabic, "Assalamu alaykum" (peace be unto you). But in fact, it meant, "Poison death be to you". The prophet's wife responded back to them with something that meant, "May Allah's wrath and curse be upon you!" The prophet, peace be upon him, corrected her by saying, "It would be sufficient to just say, "Wa alayk" (The same to you)".
This was clearly a teaching of limiting ourselves in overdoing it when replying to those who wish us evil, as nothing more than what they said back to them.