Do born-again and non-believers have the same Holy Spirit?

Answer:

"If He should determine to do so, if He should gather to Himself His spirit and His breath, all flesh would perish together, and man would return to dust" (Job 34:14-15).

There is only one Holy Spirit (Ephesians 4:4). What is being discussed in Job is not the Holy Spirit but the breath of life that God gave the world. "Then the LORD God formed man of dust from the ground, and breathed into his nostrils the breath of life; and man became a living being" (Genesis 2:7). If God took back His gift of life, Elihu's point is that nothing would survive. Our very existence is dependent on God.

"For the death that He died, He died to sin once for all; but the life that He lives, He lives to God" (Romans 6:10).

If Christ's death saves with no requirements on the recipients of the gift, then everyone is automatically saved with no need to change. But you noted that there are conditions. For example, recipients of Christ's gift are required to believe. What happens if a person stop's believing? Then clearly that person no longer qualifies for the gift. Belief isn't the only requirement. See: What Saves a Person?

A part of the problem is the assumption that once given, a gift cannot be withdrawn. "For it is impossible for those who were once enlightened, and have tasted the heavenly gift, and have become partakers of the Holy Spirit, and have tasted the good word of God and the powers of the age to come, if they fall away, to renew them again to repentance, since they crucify again for themselves the Son of God, and put Him to an open shame" (Hebrews 6:4-6). Not only is it possible, when it happens it makes it difficult to bring a person who goes completely back into the world, back to God again.

This isn't a matter of sin overpowering Christ's gift. It is a matter of the individual choosing sin over the precious gift from God.

"In Him you were also circumcised with the circumcision made without hands, by putting off the body of the sins of the flesh, by the circumcision of Christ, buried with Him in baptism, in which you also were raised with Him through faith in the working of God, who raised Him from the dead. And you, being dead in your trespasses and the uncircumcision of your flesh, He has made alive together with Him, having forgiven you all trespasses" (Colossians 2:11-13).

Paul compares baptism, which is how a person becomes born again, to circumcision in the Old Testament. See: You Must Be Born Again. Once a person became an Israelite under the Old Testament, he did not need to be circumcised again. Once a person enters the covenant with Christ in the New Testament through baptism, he does not need to be baptized again.

Entering the covenant with Christ comes with the forgiveness of sin. "Or do you not know that as many of us as were baptized into Christ Jesus were baptized into His death? Therefore we were buried with Him through baptism into death, that just as Christ was raised from the dead by the glory of the Father, even so we also should walk in newness of life. For if we have been united together in the likeness of His death, certainly we also shall be in the likeness of His resurrection, knowing this, that our old man was crucified with Him, that the body of sin might be done away with, that we should no longer be slaves of sin. For he who has died has been freed from sin" (Romans 6:3-7). While we are freed from sin, it doesn't mean sin no longer happens. "If we say that we have no sin, we are deceiving ourselves and the truth is not in us" (I John 1:8).

When sin happens, it does not mean a Christian must be baptized again. Christians are given the privilege of being able to go directly to God. "If we confess our sins, He is faithful and righteous to forgive us our sins and to cleanse us from all unrighteousness"(I John 1:9).