In the name of God, Most Merciful, Most Kind
All praise and thanks are due to God, the Lord of the worlds. We cannot find words to praise Him, as He only sees fit how laudation should be addressed to Him. We seek refuge in God from the evils within ourselves and the evils of our bad deeds. Whosoever God guides will never be led astray, and whosoever He leads astray will never find guidance. We bear witness that there is no god but God, the One without any associate, and We bear witness that Muhammad is His Messenger and bondman, peace and blessings be upon him, his kith and kin, his Companions and all those who follow their guidance until the Day of Judgment
your question: Can we greet non Muslims on their festivals?
answer: The basic principle is that, it is totally prohibited to congratulate non Muslims on their feasts because it is involves all types of forbidden things such as: shirk in Christmas, adultery(kissing and love) in Valentine's Day. And some say it is merely a word when you greet non Muslims as long as you don't believe in others's feasts. And I'm gonna try to explain my best of ability.
The reason why it is prohibited is that it is involves all types of falsehood and we cannot agree with falsehood but only with truth. We have "Festival of Sacrifice" and "Minor Festival". And the Prophet has prohibited us this. God Says:
"Prohibited to you are dead animals, blood, the flesh of swine, and that which has been dedicated to other than God, and [those animals] killed by strangling or by a violent blow or by a head-long fall or by the goring of horns, and those from which a wild animal has eaten, except what you [are able to] slaughter [before its death], and those which are sacrificed on stone altars, and [prohibited is] that you seek decision through divining arrows. That is grave disobedience. This day those who disbelieve have despaired of [defeating] your religion; so fear them not, but fear Me. This day I have perfected for you your religion and completed My favor upon you and have approved for you Islam as religion. But whoever is forced by severe hunger with no inclination to sin - then indeed, God is Forgiving and Merciful." [The Table Spread, 5:3]
So, here we learn that we should not involve in anything which God Almighty has prohibited us to do such as telling lies. God Almighty also Says:
"Or have they other deities who have ordained for them a religion to which God has not consented? But if not for the decisive word, it would have been concluded between them. And indeed, the wrongdoers will have a painful punishment" [The Consultation, 42:21]
So based on this, it is not permitted even to say to a non Muslim "happy birthday" or "happy New Year" or "Mary Christmas" or "happy holidays" or "happy Diwwali" because it involves all types of falsehood and we should not be partners in sin no matter what.
Quran says that every nation has its own festival and we have ours which is "Minor Festival" and "Festival of Sacrifice".
Anas ibn Maalik, may Allah be pleased with him, said: "The Prophet ( may Allah exalt his mention ) arrived in Medina while people were celebrating two particular festivals, so he ( may Allah exalt his mention ) asked: “What is (the significance of) these two days?” Some people replied: ‘They are days which we used to celebrate during the pre-Islamic era.’ So the Messenger ( may Allah exalt his mention ) replied: “God has replaced them for you with two days which are better, the day of ‘Eid Al-Fitr and ‘Eid Al-Adh-haa”" [Abu Daawood].
Therefore, it is not permissible for a Muslim to celebrate other than these two festivals, because is it not one of the celebrations of the Muslims.
the Prophet ( may Allah exalt his mention ): “He who imitates a people will be from among them (on the Day of Judgement).” [Abu Daawood]
CONCLUSION: So, a Muslim cannot celebrate non Muslim's festivals because it involves a lot of haram things such as: wine, singing, musical instruments, pork, shirk and falsehood and many more. But only those 2 festivals we have are the only one who has true faith. But if you are forced in this, and you have tried to escape this haram, then you are in safe area.
We ask God to guide us to the Straight Path and to make us all people of Paradise. May God send blessings and peace upon His slave and Messenger Muhammad and upon his family and companions.