There are certain passages that seem to suggest that the power of the Holy Spirit is intended for the Church, the Body of Christ. If this is the case, and if we define Christian as someone who belongs to the Body of Christ, then it would follow that non-Christians do not have access to this power (or at least not to the same extent as in a Christian, for whom the power is intended).
1 Corinthians 12 (ESV):
Now concerning spiritual gifts, brothers, I do not want you to be uninformed. 2 You know that when you were pagans you were led astray to mute idols, however you were led. 3 Therefore I want you to understand that no one speaking in the Spirit of God ever says “Jesus is accursed!” and no one can say “Jesus is Lord” except in the Holy Spirit.
4 Now there are varieties of gifts, but the same Spirit; 5 and there are varieties of service, but the same Lord; 6 and there are varieties of activities, but it is the same God who empowers them all in everyone. 7 To each is given the manifestation of the Spirit for the common good. 8 For to one is given through the Spirit the utterance of wisdom, and to another the utterance of knowledge according to the same Spirit, 9 to another faith by the same Spirit, to another gifts of healing by the one Spirit, 10 to another the working of miracles, to another prophecy, to another the ability to distinguish between spirits, to another various kinds of tongues, to another the interpretation of tongues. 11 All these are empowered by one and the same Spirit, who apportions to each one individually as he wills.
12 For just as the body is one and has many members, and all the members of the body, though many, are one body, so it is with Christ. 13 For in one Spirit we were all baptized into one body—Jews or Greeks, slaves or free—and all were made to drink of one Spirit.
27 Now you are the body of Christ and individually members of it. 28 And God has appointed in the church first apostles, second prophets, third teachers, then miracles, then gifts of healing, helping, administrating, and various kinds of tongues. 29 Are all apostles? Are all prophets? Are all teachers? Do all work miracles? 30 Do all possess gifts of healing? Do all speak with tongues? Do all interpret? 31 But earnestly desire the higher gifts.
And I will show you a still more excellent way.
See also Acts 1:8 (notice the relationship between the power of the Holy Spirit and being a witness of Jesus Christ):
But you will receive power when the Holy Spirit has come upon you, and you will be my witnesses in Jerusalem and in all Judea and Samaria, and to the end of the earth.” [ESV]
Additionally, I find this answer by Mike very insightful:
For the most part all Christianity hold to the doctrine of original
sin. This means that human nature was cursed in Adam and subject to a
total sinfulness under the law, separate from God, with no true good
or love remaining. (Rom 5:12)
“There is no one righteous, not even one; there is no one who
understands; there is no one who seeks God. All have turned away, they
have together become worthless; there is no one who does good, not
even one.” Rom 3:11-12 NIV
In this state of being a ‘sinner’ nothing good exists as it was meant
to be. (Rom 7:18)
Therefore sinners do not have the Spirit as they are without God.
(Rom 8:9) This means any quality of their nature that might resemble a
fruit of the Spirit, is not from the Spirit but from the flesh. So
in simple terms a non Christian has zero fruits of the Spirit. The
fruits of the Spirit presume faith in Christ as a priori for that
Spirit of Christ to be working those fruits in the sanctified soul.
In deed any pretended love must be love for God ‘by faith in Christ’
otherwise it is not love and anything that is not from faith is sin.