At maximum, people think that Muslims are not allowed to wish them on their non-Muslim festivals. Unfortunately, the situation is much more dangerous than this. 

Sharia Teachings are following this pathway:

* Non-Muslim festivals are Evil and they are from Devil/Satan. 

* Not only Muslims are not allowed to wish the non-Muslims, but also Muslims have to keep "hatred" against non-Muslim festivals. 

* And in the last stage, when Muslims get enough power, then they should by force prohibit the non-Muslims from openly celebrating their festivals. 

The largest Saudi Salafi Fatwa website writes (link):

Greeting Christians on the occasion of their festivals: 

What is forbidden with regard to congratulating the Christians on the occasion of their festivals is expressing joy to them, being overly polite and showing approval of their actions, even if that is only expressed outwardly without feeling it inwardly. 

The prohibition concerns those who show any kind of participation or approval, such as giving gifts, verbal greetings (i.e. saying merry Christmas), taking time off work, making food, going to places of leisure, and other customs of celebration. Having any intention other than what is said (when congratulating them) does not make it permissible. The outward appearance of these actions is sufficient to say that it is prohibited. 

It is well known that most of those who take a lenient attitude towards such matters do not intend to join the Christians in their shirk; rather what motivates them in some cases is showing politeness, and in other cases it is shyness, but politeness with regard to falsehood is not permissible; rather what is required is to denounce the evil and strive to change it. 

In fact some of the (early) scholars are of the view that the one who does these things is a kaafir, because it involves venerating the symbols of kufr. 

Sahabi (companion) ‘Abdullah ibn ‘Amr ibn al-‘Aas said: The one who follows the tradition of non-Muslims and observes their Nawrooz and Mahrajaan (festivals), and imitates them until he dies like that, will be gathered with them on the Day of Resurrection. 

(There is an Ijma' that) Ameer al-Mu’mineen ‘Umar ibn al-Khattaab, the Sahaabah and all the imams of the Muslims stipulated that non-Muslims should not celebrate their festivals OPENLY in the lands of the Muslims; rather they should do that privately in their house.

On every Eid day, please do wish happy Eid to the moderate Muslims (as they still have humanity in them and they don't follow the extremist Mullah or extremist Islam), but please also try to make people aware of this danger. 

Please don't take it lightly.  

At moment Muslims are weak and they only prohibit their children to wish the non-Muslims and tell them to keep hatred against their festivals in their hearts. 

But as soon as Muslims get enough power, then in the next step, they are going to prohibit the non-Muslims from even celebrating their festivals while all these festivals are from the Devil.  

For example, we already have a Ban on Valentine's Day in Pakistan. And this ban is not from the extremist Muslims, but it is from the Government of Pakistan and from the High Court of Islamabad. 

PS: Muslim claim that Atheists don't have any festival of their own

Muslims told me on Eid (a few years ago), Atheists have no festivals of their own as it is not a system but a disease, and thus Ex-Muslims atheists still celebrate Eid and Ex-Christian atheists still celebrate Christmas.  

I told them, people celebrate due to their human nature, and not due to religion. 

In many countries and cultures, the start of spring is celebrated as a festival. For example, Iranians have the Norouz, which was again related to starting of the spring season.  Chinese also celebrate the start of the spring season. 

Then humanity started the celebrations of "mother's day" and "father's day" which are based on love and totally non-religious festivals.  

Then humanity started the celebration of "Valentine's Day", which is again not a religious festival but based upon love and human nature. 

There are tons of celebrations in schools and colleges and universities and all of them are non-religious. 

There are tons of traditional celebrations in every part of the world and they are also non-religious in nature. 

Humans don't need Allah to celebrate the festivals, but we need to follow our human nature and use our brains to get rid of Allah's monopoly on the festivals.