Some do, some do not. However, it should be understood that they have the same level of "Spirit of GOd’ as a rightous gentil would in the Bible. That is to say, not like a baptised Christian or even a Jew.
Are you saying that the Catholic Church teaches that non-Christians can have the Spirit of God? I believe the Bible clearly teaches that we receive the Spirt of God upon faith in Christ as a deposit of faith. Please help me understand because the Spirt of God is just another name of God the Holy Spirit, or Spirit of Christ.
Ephesians 1
Spiritual Blessings in Christ
3 Blessed be the God and Father of our Lord Jesus Christ, who has blessed us in Christ with every spiritual blessing in the heavenly places, 4 even as he chose us in him before the foundation of the world, that we should be holy and blameless before him. In love 5 he predestined us [2] for adoption as sons through Jesus Christ, according to the purpose of his will, 6 to the praise of his glorious grace, with which he has blessed us in the Beloved. 7 In him we have redemption through his blood, the forgiveness of our trespasses, according to the riches of his grace, 8 which he lavished upon us, in all wisdom and insight 9 making known [3] to us the mystery of his will, according to his purpose, which he set forth in Christ 10 as a plan for the fullness of time, to unite all things in him, things in heaven and things on earth.
In him we have obtained an inheritance, having been predestined according to the purpose of him who works all things according to the counsel of his will, so that we who were the first to hope in Christ might be to the praise of his glory. In him you also, when you heard the word of truth, the gospel of your salvation, and believed in him, were sealed with the promised Holy Spirit, who is the guarantee of our inheritance until we acquire possession of it, to the praise of his glory.
Romans 5
Therefore, since we have been justified by faith, we have peace with God through our Lord Jesus Christ. Through him we have also obtained access by faith** into this grace in which we stand, and we rejoice in hope of the glory of God. More than that, we rejoice in our sufferings, knowing that suffering produces endurance, and endurance produces character, and character produces hope, and hope does not put us to shame, because God’s love has been poured into our hearts through the Holy Spirit who has been given to us.**