Here is a response I am going to provide to counter-missionary Tovia Singer and I hope to respond to the arguments that he lays out in his article to the best of my ability.
You can find Singer's original article here: http://outreachjudaism.org/jesus-passover/
"Evangelical Christians often draw a comparison between the Paschal Lamb and Jesus, insisting that the former foreshadows the latter. This idea is advanced in the New Testament, particularly in the fourth Gospel, where John portrayed Jesus as the fulfillment of the Passover lamb. Yet how valid a point is this? What is the meaning of this holiday sacrifice? Is there a relationship between this festival offering and atonement for sin?
The Bible relates in Exodus 12:3-13 that as the Jewish people were preparing themselves for the momentous Exodus from Egypt, God commanded them to slaughter a year-old sheep or goat on the 14th day of the first month (Nissan). They were to place its blood on the outside doorposts of their homes. Because Christians insist that the blood of the Paschal lamb foreshadowed the atonement of the blood of Jesus at Calvary, it behooves us to question the soundness of this claim.
The Passover lamb did not atone for sin and accordingly, this idea is nowhere to be found in the Jewish Scriptures. It goes without saying that the notion that the Paschal Lamb is a representation of a crucified savior or an atonement is alien to the teachings of the Torah and is not even mentioned by the first three Gospels."
Just because the first three Gospels don't mention the subject of Jesus being the Passover Lamb, this wouldn't indicate that it is a teaching that is not found in the NT.
Regarding as to why Jesus is described as the Passover lamb in the NT is quite simple, Even though the Exodus story doesn't mention repentance, that isn't what Paul is conveying when he alludes to the Passover.
His whole point is that just as the blood was used to cover the doorpost and cause the angel of death to Passover the Israelites, so the blood of Christ which we are washed in, causes God's wrath to Passover us. Paul does mention putting away the old leaven (evil and sin), but his point, again, the blood on the door post and the blood of Jesus acting as a protective covering from God's wrath.
"A mindful study of the Jewish Scriptures reveals that the Paschal Lamb was alluded to long before the Exodus from Egypt. Centuries earlier, Abraham’s faith was tested by God when he commanded him to sacrifice his beloved son Isaac. Genesis 22:7-8 relates that as the two ascended Mount Moriah together, Isaac asked his father,
“Here is the fire and the wood, but where is the lamb for the offering?” Abraham then replied, “God will see to a lamb for an offering, my son.”
The question that comes to mind is, what happened to that lamb that Abraham promised? A few verses later we find that ram was sacrificed rather than a lamb! Where was the lamb to which Abraham was prophetically referring?
The answer of course is that our father Abraham was prophetically alluding to the Paschal lamb. Just as God tested Abraham’s faith to demonstrate his worthiness to be the father of the chosen people, the young Jewish nation also had to have their faith tested to show their worthiness to participate in the exodus from Egypt, receive the Torah at Mount Sinai, and emerge as the progenitors of the covenant people who would forever be known as “a light to the nations.”"
The point about what was being sacrificed misses the point about why Christians appeal to the passage in the first place, the lamb or ram being sacrificed is a minute point that overlooks what Christians claim about the passage.
The point that is made is that God will provide as substitute in the future to deal with our sin. Isaac is about to be sacrificed and God stops him, providing instead a ram for him to be sacrificed in Isaac's place. Bringing up the objection of being a burnt offering or not a lamb is just nitpicking when it comes to typological fulfillment.
Though I find Singer's point about Abraham prophetically alluding to the Passover Lamb an interesting point.
"During the period of the Exodus in Ancient Egypt, the lamb was deified and worshiped as a god. By Egyptian law, it was therefore forbidden to harm a lamb in any way; such an act was considered a crime punishable by death.
For this reason, Moses refused Pharaoh’s initial offer that the Jews bring their sacrifice to God while remaining in Egypt, following the third plague of lice. Moses explained to Pharaoh that it would be impossible for his people to sacrifice these animals in this land because the Egyptians would execute us for carrying out this ceremony (Exodus 8:25-26).
The Almighty, therefore, tested the faithfulness of the Jewish people by commanding them to kill Egypt’s cherished god, and place the lamb’s blood on their doorposts, displayed for all of their neighbors to see. Only those Israelites who, like Abraham, demonstrated that they feared nothing but the God of Israel were deemed worthy to have their homes “passed over” during the tenth and final plague.
It is worth noting that the synoptic gospels, i.e. the gospels of Matthew, Mark, and Luke, do not associate Jesus with the Paschal Lamb. The Book of John, on the other hand, draws a clear link between the two (John 1:29-34). The synoptic Gospels insist that Jesus was crucified on the first day of Passover – the 15th day of Nissan. Written several decades after the synoptic Gospels, th John’s author accordingly has Jesus crucified on the eve of Passover, the 14th day of Nissan, when the lambs were slaughtered. As a result, the Passover Seder is noticeably absent in John’s Passion Narrative."
Though a link is drawn between the passover lamb and Jesus is made, The NT NEVER claims that Jesus had some connection to any deity in Egypt and such a thought was never in their mind when the comparisons were made.
I have already mentioned why Paul made the connection to the Passover Lamb with Jesus so I needn't go over that again, but even if I grant Singer's point as valid that the Israelite's faithfulness was tested by killing the Egyptian's most cherished god, none of the NT writers had Egyptian theology in mind or something from that. See my article "The Lamb of God": http://answering-judaism.blogspot.co.uk/2014/02/the-lamb-of-god.html
Just because certain details are not mentioned in a Gospel, that doesn't make a point a particular Gospel makes to be false. The Gospels are each written to a specific audience at a given time. Hence, they would only mention things that are relevant to the point that they themselves are trying to convey. The same principle applies to the virgin birth Matthew and to the I AM statement of Jesus in John, they were points only made to the audience they were addressing specifically.
Any information I have missed I will be happy to look into if the Lord Wills.