I am not sure how to really explain the problem with this statement save from the self evident truth that the Holy Spirit is not 'in the flesh'.
The word incarnation means at the literal basic level to become flesh or to become meat. In the Nicene Creed to we say "and was incarnate of the Holy Spirit and the Virgin Mary, and became man." or in the ELLC text "was incarnate of the Holy Spirit and the Virgin Mary and became truly human". In no sense has anyone, to my knowledge, suggested that the Holy Spirit became flesh.
The Incarnation is part of the unique mission of God to reconcile humankind to himself. I think you would find Anselm's book Cur Deus Homo especially helpful in coming to grips with this important matter.
Given that the Holy Spirit is not human, but divine, it makes no sense, to me at least, in terms of language or theology to speak of the incarnation of the Holy Spirit, but rather denigrates the work of the Father, the Son and the Holy Spirit in the incarnation of Jesus Christ.