Is Saying `Eid Mubarak Bi`dah?
Question

What is the ruling on offering congratulations on the occasion of ` `Eid? What is the ruling on shaking hands and embracing after the ` `Eid prayer?
Praise be to Allah.It was narrated that the Companions (may Allah be pleased with them) used to congratulate one another on the occasion of `Eid by saying Taqabbal Allahu minna wa minkum (May Allah accept (this worship) from us and from you).
It was narrated that Jubayr ibn Nufayr (may Allah have mercy on him) said: When the companions of the Messenger of Allah (peace and blessings of Allah be upon him) met one another on the day of `Eid , they would say to one another: “Taqabbal Allahu minna wa mink (May Allah accept (this worship) from us and from you).” Al-Hafidh (may Allah have mercy on him) said: “Its chain of narration is good.” 
Imam Ahmad (may Allah have mercy on him) said: “There is nothing wrong with one man saying to another on the day of `Eid: Taqabbal Allahu minna wa mink (May Allah accept (this worship) from us and from you).” This was narrated by Ibn Qudamah in Al-Mughni. 
Shaykh Al-Islam Ibn Taymiyah (may Allah have mercy on him) was asked in Al-Fatawa Al-Kubra (2/228): Does the common `Eid greeting that is on people`s lips – “ `Eid Mubarak” etc – have any basis in Shari`ah or not? If there is a basis for that in Shari`ah, what should we say? 
He (may Allah have mercy on him) replied: 
“With regard to the greeting on the day of `Eid , which people say to one another when meeting after the prayer , “Taqabbal Allahu minna wa minkum wa ahalahu Allah `alayka (May Allah accept (this worship) from us and from you and may you live to see another `Eid)” etc, this was narrated from a number of the Companions who used to do that and allow others to do so too, and from the Imams such as Imam Ahmad. But Ahmad said: I do not initiate this greeting with anyone. But if someone greets me in this manner I return his greeting. That is because returning a greeting is obligatory, but initiating this greeting is not a Sunnah that is enjoined, but neither is it forbidden. The one who does it has an example (whom he imitated) and the one who does not do it also has an example (whom he imitated). And Allah knows best.” 
Shaykh Ibn `Uthaymin (may Allah have mercy on him) was asked: What is the ruling on offering `Eid greetings and is there a particular wording to be used? 
He replied: 
“It is permissible to offer greetings and congratulations on `Eid, and there is no specific greeting. Rather the greetings that people customarily use are permissible so long as no sin is involved.” 
He (may Allah have mercy on him) also said: 
“Some of the Companions offered greetings and congratulations on the occasion of `Eid. Even if we assume that they did not do that, it has now become something customary that people are used to doing, congratulating one another on the occasion of `Eid and on completing the fast and Qiyam.” 
And (may Allah have mercy on him) he was asked: what is the ruling on shaking hands, embracing and congratulating one another after the `Eid prayer? 
He replied: 
“There is nothing wrong with these things, because people do not do these things as acts of worship intended to draw them closer to Allah, rather they do them because they are customary, and to honour and show respect to one another. So long as there is nothing in Shari`ah to indicate that a custom is forbidden, then the basic principle is that it is permissible.” (Majmu` Fatawa Ibn `Uthaymin, 16/208-210)
And Allah knows best.