Has the term "born again" always been synonymous with baptism with the Christians of the Early Church? Has the term "born again" always been synonymous with baptism with the Christians of the Early Church?

St. Irenaeus (202) wrote the following in 185, which many claim it refers to infant baptism:


  Irenaeus, c. A.D. 185:
  
  He came to save all through means of Himself—all … who through Him are born again to God—infants, and children, and boys, and youths, and old men. He therefore passed through every age, becoming an infant for infants, thus sanctifying infants; a child for children, thus sanctifying those who are of this age, being at the same time made to them an example of piety, righteousness, and submission … (Against Heresies II:22:4)


The article on infant baptism from this site claims that the term "born again" was synonymous with baptism to early Christians. 


  Note here that the term "born again" was synonymous with baptism to early Christians. That really didn't change until the time of the pietists in the 17th century.


Is there any historical evidence to support the contrary of this statement in the Early Church or the Church Fathers?
Some Clarification

You question specifically asks if there were any in the Early Church (~2nd to 4th century) who separated "born again" with baptism. Meanwhile, the site you quote claims that this separation took place in the 17th century. Because your question is only concerned with the Early Church, I will not consider any of the possible historical data from the 5th to 17th century.

It is also a bit dangerous to say anyone regards "born again" as being synonymous with baptism. They are two difference terms which have slightly difference meanings--even to those believing in baptismal regeneration. It is similar to how "regeneration" means something slightly different than "born again" despite the fact that most Christians regard them as mostly describing the same event.

Therefore, I take it that when you ask if early Christians all thought that "born again" was synonymous with baptism that you mean to ask if early Christians all thought that "born again" generally describes the event of baptism.

Perhaps a better way to have asked the question would been to ask if there were any early Christians who did not believe in baptismal regeneration. But no worries, I will still answer the question in the terms you've used.

The Answer

The short answer to your question is that there is absolutely no historical data from the Early Church demonstrating that any early Christians understood "born again" as being significantly distinct from baptism. On the contrary, there are a plethora of quotes from the Early Church demonstrating that these Christians understood the two terms to be nearly synonymous. Here are a few.

Theophilus of Antioch in 181 CE


  Moreover, those things which were created from the waters were blessed
  by God, so that this might also be a sign that men would at a future
  time receive repentance and remission of sins through water and the
  bath of regeneration -- all who proceed to the truth and are born
  again and receive a blessing from God. (To Autolycus 2:16)


Irenaeus in 190 CE


  "And [Naaman] dipped himself...seven times in the Jordan" [2 Kings
  5:14]. It was not for nothing that Naaman of old, when suffering from
  leprosy, was purified upon his being baptized, but [this served] as an
  indication to us. For as we are lepers in sin, we are made clean, by
  means of the sacred water and the invocation of the Lord, from our old
  transgressions, being spiritually regenerated as new-born babes, even
  as the Lord has declared: "Except a man be born again through water
  and the Spirit, he shall not enter into the kingdom of heaven."
  (Fragment 34)


Recognitions of Clement 221 CE. Notice in this quote that the author mis-remembers John 3:5 leaving out the critical "and of the Spirit."


  But you will perhaps say, "What does the baptism of water contribute toward
  the worship of God?" In the first place, because that which has
  pleased God is fulfilled. In the second place, because when you are
  regenerated and born again of water and of God, the frailty of your
  former birth, which you have through men, is cut off, and so ...you
  shall be able to attain salvation; but otherwise it is impossible. For
  thus has the true Prophet [Jesus] testified to us with an oath:
  "Verily, I say to you, that unless a man is born again of water [sic] he
  shall not enter into the kingdom of heaven." (Recognitions 6:9)


Cyprian of Carthage 200-258 CE


  [When] they receive also the Baptism of the Church...then finally can they be fully
  sanctified and be the sons of God...since it is written, "Except a man
  be born again of water and of the Spirit, he cannot enter into the
  kingdom of God." (Letters 71[72]:1)


Conclusion

There are many more quotes I could fish up for you, but I believe I have given you enough of a starting point. While we could never be historically certain that no early Christians thought "born again" and baptism were significantly distinct, it is indeed very odd that there are no surviving documents demonstrating their existence.