Fortunately no. Neither is there any need for any Christian equivalent of the hadith. The nearest the Christian faith has to the hadith are the Gospel accounts of Matthew, Mark, Luke, and John. Each one of these four accounts is considered as Scripture, fully given by inspiration of God. And each one is easy to understand. Each one stands alone as an account of the ministry of our Lord Jesus Christ. It is true that if you have some familiarity with the Old Testament then, at times, you will better appreciate what is written in a gospel account, but such knowledge is not needed to grasp the essential message of any one of the gospel accounts.
The four gospel accounts, then, are the Christian's hadith. But no, they are better, far better, than the hadith, because the Christian faith says that they are given by inspiration of God, and the hadith are never claimed within the Islamic faith to be given by inspiration of Allah.
Because the Christian has these four accounts given by inspiration of God why would he need anything else? God has determined exactly what is needed for anyone to believe, it would be foolhardy to add to his writings.
On the other hand, the writings of Islam must be considered very peculiar to every Christian, and indeed to any neutral bystander as well. Let me explain:
According to the Islamic faith the Quran is the only fully inspired word of Allah. Now the Quran is quite short, being only about four fifths the length of the New Testament. The Quran is also not always easy to follow. Some things are written which can hardly be understood unless you know the historical context in which they were written. And the problem is the historical context is not always declared within the Quran: you have to go to the hadith to find the context, and so to understand the Quran. This seems very peculiar to me as a Christian: why would God not give enough information in his word? Why would he rely on the mere uninspired writings of men before his word can be understood?
If the hadith are not given by inspiration of God then how much can we trust them? Islam accepts this is a problem as well, and has produced a list of the hadith in order of supposed reliability. I do not know how many compilations of hadith there are in total, but the first six in terms of supposed reliability are Sahih Bukhari, Sahih Muslim, Sunan Abu Daud, Sunan Tirmidhi, Sunan ibn Majah, and Sunan Nisai. If something is found in the most reliable of them all, the Sahih Bukhari, then it is possibly likely true, but a Muslim is not required to believe it, he can still doubt it.. after all the Sahih Bukhari is not the word of God. But if the same account is found in several of these six books or other of the hadith then if the Muslim still doubts then the very foundations of the Islamic faith are slipping through his fingers like sand and there's not going to be much else to hold on to.
Those of us who believe the Bible also believe that the first three gospel accounts were written before 70 AD when the Temple of Jerusalem was destroyed by the Romans, and that John's Gospel was written about 70 to 90 AD by the Apostle John. We believe Matthew's Gospel was written by the Apostle Matthew, Mark wrote down what Peter preached (according to the early writer Papias), and that the historian and doctor, Luke, carefully interviewed eyewitnesses to supplement what he took from either Mark or Matthew. All four gospels then were written within a generation of the life of Christ, either by eyewitnesses themselves, or by those interviewing eyewitnesses.
What about the hadith? What does the Muslim believe about them? When do Muslims say they were written and how did they come to be? The Muslim believes that the hadith were passed on by word of mouth from one generation to the next, through reliable people, until they were finally written down between two hundred and two hundred and fifty years after the death of the founder of Islam, Mohammad. Let me be brutally honest: as a Christian what the Muslim is asking me to believe is that God gave a book, the Quran, which cannot be always understood, so Muslim men commit their trust for the understanding of this book, to other books, which God did not inspire, and which are the result of a game of Chinese Whispers which went on for at least two hundred years. And this is supposed to be a matter of heaven or hell for all eternity? I'm sorry, I think its completely unacceptable, and unbelievable. Allah is repeatedly called "the Compassionate, the Merciful" - to entrust the salvation of men, women and children to such a vague foundation as this is surely not good enough for a merciful, compassionate God. I accept God is Compassionate and Merciful, and so I do not think this can possibly be his handiwork.
So does the Christian have an equivalent to the Muslim hadith? Praise God, NO! He has the gospel accounts, which are clear and sufficient in themselves, and are written to inspire faith in Christ as the Saviour of the world, Jews, and gentiles including any Muslim who will put their trust in Christ as the Son of God. As John in his gospel says at the start of his gospel account
In the beginning was the Word, and the Word was with God, and the Word was God. The same was in the beginning with God. All things were made by him and without him was not any thing made that was made... And the Word was made flesh, and dwelt among us" (John 1:1-3 and John 1:14)
and near the end of his gospel account John wrote:
"Many other signs truly did Jesus in the presence of his disciples which are not written in this book; but these are written, that you might believe that Jesus is the Christ, the Son of God; and that believing you might have life through his name" (John's Gospel 20:30,31).
Believe on Christ, Muslim friend.