There are several times in John's gospel where this phrase about being "born again" is stated, and all prior to the verse in question, which does not actually use that phrase. If we get a clear grasp of what being born again means, then it should not be difficult to know whether it's the same event being spoken of in John 5:24.
Jesus first said it, telling Nicodemus that nobody could even see the Kingdom of God (let alone enter into it) unless they were "born from above" (John 3:3 YLT). This phrase is then variously enlarged upon by other statements in that conversation:
"...born of water and of the Spirit..." (v. 5)
"that which is born of the Spirit is spirit" (v. 6)
"...thou hast not known when he [the Spirit] cometh, and whether he goeth; thus is every one who hath been born of the Spirit." (v. 8)
Nicodemus would know that Jesus was referring to the water baptism of John the Baptist as this being 'born' of water. That is why those baptised of John were then prepared to receive Jesus as the Messiah. That is why Jesus said, "We speak that we do know, and testify that we have seen". "We"? Yes, John's testimony about Jesus being the One from heaven, and Jesus' own testimony. They are to be believed. Jesus next went on to say something quite mysterious:
"And as Moses did lift up the serpent in the wilderness, so it
behoveth the Son of Man to be lifted up, that every one who is
believing in him may not perish, but may have life age-during, for God did so love the world, that His Son - the only begotten - He gave, that every one who is believing in him may not perish, but may have life age-during." John 3:14-16 (YLT)
Then, John the Baptist is questioned as to why he is baptising less people than Jesus' disciples are. John is very happy about that, saying, "A man is not able to receive anything, if it may not have been given him from the heaven" (v. 27). John keeps pointing to Christ, concluding, "He who is believing in the Son, hath life age-during, and he who is not believing the Son, shall not see life, but the wrath of God doth remain upon him" (v. 36). So ends chapter 3.
Chapter 4 is a different location, there is a different conversation with a woman at a well, Jesus speaking of giving living water that springs into everlasting life, and the hour having come when true worshippers will worship the Father in Spirit and truth. It is impossible for anyone to worship God as Father unless they have been born after the Spirit from above. Some days later the Samaritans in that village had believed Jesus and were born again.
Chapter 5 starts, the location now being Jerusalem. Jesus spoke about the Father raising the dead (v. 21), that God has committed all judgment to the Son (v. 22) hence the honour Christ is due (v. 23), and now the verse in question, those who hear Christ's word and believe the Father in heaven who sent him have everlasting life, not being condemned for they have passed from death to life.
But verse 24 cannot stand apart from the preceding verses and verse 25 — that the hour had now come (while Jesus was on Earth) when the dead would hear the voice of the Son of God and live. This is the first resurrection, inward and spiritual; pertaining not to sight but to faith, which is what verse 24 is all about.
The life Jesus spoke of was spiritual life that never ends; that begins with the unseen movement of the Holy Spirit bringing a spiritually dead person to newness of life so that they believe Jesus and the Father.
Significantly, the rest of chapter 5 has Jesus pointing back to the witness of John the Baptist, and to the continuing witness of the Father. Those who come to Christ will have life, as the Scriptures state.
Jesus himself said (verses 39-40) that those who come to him will have life. They are already living physically, but they need the unction of the Holy Spirit from on high to receive Christ by faith so as to live eternally as believers born from above. "A man is not able to receive anything, if it may not have been given him from the heaven" (John 3:27). All that is in chapter 5 fits in beautifully with all that is in chapter 3. Both chapters are speaking about being born from above but by chapter 5 Jesus has explained more, linking being born from above with the first resurrection.