This is an incredibly common and frustrating bit of bad theology. There's no textual or archeological basis for it, and the really interesting mystery is where exactly this myth even came from.
I, too, used to be a “preachermon”. I never found evidence that this argument was anything but apocryphal.
It would make a better argument to find a text describing how Jesus referred to a rich person (real or parable) and said, “this rich person gets it, be like him”. Direct, and without the mental backflips.
it is, however, very easy to find other ways jesus has phrased the idea "the rich don't get into heaven". In matthew he says the last shall be first and the first shall be last, in acts he has his disciples sell all of their stuff and pool the money, and in several other places he tells the faithful to give all of their stuff away.
none of this is at all true, none of it is supported by the historical record, none of it is supported by the rest of jesus's teachings (which explicitly and repeatedly state that the rich are not welcome in heaven), there never was an "eye of the needle" gate, no one even posited the existence of such a gate until the 9th century CE and the other gospels use different phrasing of "a camel passing through the eye of a needle" that indicate that "the eye of a needle" isn't a proper noun referring to a singular entity with a commonly-known name.
This is literally not true by scripture; Abraham was very rich, and is considered righteous because of his faith. He did not withhold even his own son from God. Money did not own Abraham's faith.
Job refused to curse or condemn God even when he lost most of his family and all of his holdings - his friends tried to tell him that because he lost his riches, he had obviously sinned, but he refused this. He gained back the things he lost, because of his faith in God.
Job 1:20-22
20 Then Job stood up, tore his robe, shaved his head, fell to the ground, bowed very low, 21 and exclaimed:
“I left my mother’s womb naked,
and I will return to God naked.
The Lord has given,
and the Lord has taken.
May the name of the Lord be blessed.”
22 Job neither sinned nor charged God with wrongdoing in all of this.
In those times, rich people were considered blessed by God, poor people or those afflicted with disease were considered cursed by God. People afflicted from birth were said to have been "born in sin" due to the sins of their parents.
The Pharisees and Sadducees were wealthly, influential people who preached exactly this. The Sadducees in particular didn't believe in an afterlife, and so were focused on only the "here and now" and material things of this world. Jesus specifically called them out to let them know their wealth wouldn't get them into heaven, and their success was not a sign of righteousness.
Jesus distinctly preached that money could not buy salvation, and that those whose focus was on money could not focus on God, and would therefore be condemned. He explicitly called out Zaccheus, a tax collector, when Zaccheus promised to repay any money he'd taken in bad faith four times over and to give away half of what he owned to the poor:
Luke 19:9-10:
9 Jesus said to him,"Today salvation has come to this house, because this man, too, is a son of Abraham.
10 For the Son of Man came to seek and to save the lost."
Zaccheus probably had a lot of money left even when he was done, but the point is that money was no longer the priority in his life.
God may choose to bless people with prosperity, but your wallet doesn't make you righteous. It doesn't make you unrighteous either - your actions and your faith, or lack thereof condemn you. The whole of the Law is:
Matthew 22:34-40
34 When the Pharisees learned that he had silenced the Sadducees, they gathered together, 35 and, to test him, one of them, a lawyer, asked this question, 36 “Teacher, which is the greatest commandment in the Law?”
37 Jesus said to him, “ ‘You shall love the Lord your God with all your heart, and with all your soul, and with all your mind.’ 38 This is the greatest and the first commandment. 39 The second is like it: ‘You shall love your neighbor as yourself.’ 40 Everything in the Law and the Prophets depends on these two commandments.”
The "eye of the needle" was a (very small) gate into Jerusalem.
To get a camel through that gate, it has to lower its head and crawl on its knees.
So Jesus was calling rich people camels; camels can be very arrogant beasts so it fits.