Do any of the Church Fathers draw a connection between Eve and the Holy Spirit? In Wild at Heart, author John Eldredge takes some effort to describe the language used in Genesis 2:18 to refer to Eve. While some translations and commentors translate the hebrew "עֵ֖זֶר כְּנֶגְדּֽוֹ׃" (ezer kenegdo) "suitable helper" or "help mate", Eldredge comments on the weight of that description while discussing the sanctity and gravity of the marriage union, 


  It means something far more than just "helper"; it means "lifesaver." The phrase is only used elsewhere of God, when you need him to come through for you desperately. "There is no one like the God of Jesherun, who rides on the heavens to help you" (Deut. 33:26). Eve is a life giver; she is Adam's ally. 


While meditating on this, I was reminded of what Jesus said to the disciples before his crucifixion:


  I will ask the Father, and He will give you another Helper, that He may be with you forever; (John 14:16, NASB)


Do any of the Church fathers compare Eve and the Paraclete?
Perhaps not surprisingly, in light of Ephesians 5, church fathers widely connect Eve with Christ's Church.  This seems to displace other interpretations, like a connection between Eve and the Holy Spirit.  However, it is at least hinted at in Chrysostom, when he compares the Old Testament man to the New Testament man: 


  Then [i.e., OT times] He said, “Let us make for him a help,” but here [i.e., NT times] He said nothing of the kind. What other help shall he need, who has received the gift of the Spirit? What further need of assistance has he, who belongs to the Body of Christ? (Homilies on John, 25.2)


That is, in the OT the "help" God provided to Adam was Eve, but in the NT, he has already provided the "gift of the Spirit," so no additional help is required.

Granted, the connection is weakened because Chrysostom goes on to describe other ways in which the New Testament believer has already been provided with "help." 
Nonetheless, it's clear that Chrysostom connects the "help" for Adam of Genesis 2:18 (Eve) with the "gift of the Spirit" for the New Testament believer.