Buddhism and Christianity both enjoin what I will call moral self-denial. But Buddhism is more radical in that it connects moral self-denial with metaphysical self-denial. Thus Buddhism denies the very existence of the self, whereas Christianity in its orthodox versions presupposes the existence of the self: Christian self-purification falls short of eliminativism about the self. Nevertheless, there are points of comparison between the 'No Self' doctrine of Buddhism and the Christian doctrine of the self.  Just as the full appreciation of the mother tongue comes only to those who study foreign languages, the  full appreciation of the 'mother religion' comes only to those who study foreign religions.

In his Combattimento Spirituale (1589), Lorenzo Scupoli writes:

You my mind, are not mine: you were given me by God. Neither are the powers active within me — will, with its energy — mine. Nor does my feeling, my ability to enjoy life and all my surroundings belong to me. My body with all its functions and requirements, which determine our physical well-being, is not mine either . . . . And I myself belong not to me, but to God. (Unseen Warfare, St. Vladimir's Seminary Press, 1995, p. 172)

I'll hazard a gloss: I am not the master of my fate nor the captain of my soul. Everything I have I have from God who created me ex nihilo and sustains me  moment-by-moment. I am not my own man. I do not have property-rights in my body nor in any attribute or adjunct of what I take to be my self. For this very reason, suicide is a grave sin. If a substance is anything metaphysically capable of independent existence, then I am not a substance: only God is a substance in the plenary sense of the term.  

Apart from the references to God, this meditation of Scupoli, of which the above is merely an excerpt, bears a striking resemblance to the  Anattalakkhana Sutta. Buddha there examines each of the khandas, body, feeling, perception, etc., and concludes with respect to each of them that "This is not mine. This is not my self. This is not what I am." In Scupoli we encounter virtually the same litany: body, feeling, mind . . . of each of which it is true that "This is not mine," etc.  Of course, nothing depends on the exact taxonomy of the khandas or personality-constituents. The point is that however one classifies them, no one of them, nor any combination of them, is veridically identifiable as one's very self. I say 'veridically,' since we do as a matter of fact falsely identify ourselves with all manner of item both within us (feelings, memories, etc.) and without us (property, progeny, etc.) My house, my child, my brilliant insights.  A theologian might identify himself with his theology, when he must know that his theology could be light-years away from God's theology (God's self-knowledge).

These false self-identifications are part of what our ignorance/sinfulness consists in. (The forward slash in my typography stands for inclusive disjunction, the inclusive 'or.')

Thus Scupoli (whom I take to be a representative Christian, and who is of interest here only as such) and Buddha agree with respect to the (negative) thesis that nothing in one's outer or inner experience is veridically identifiable as one's very self. Thus nothing that we ordinarily take to be ourselves (our bodies, our thoughts, feelings, memories, etc.) can in truth be one's self. But there is also similarity in their reasoning. One way Buddha reasons is as follows.

If x (body, feeling, perception, etc.) were my very self, or were something that belonged to me, then I would have complete control over x. But it is evident that nothing is such that I have complete control over it. Therefore, no x is either my self or anything that belongs to me. This could be called the 'complete control argument' against the existence of the self.  Scupoli has something similar:

Let us remember that we can boast only of something which is a direct result of our own will and is done by us independently of anything else. But look how our actions proceed. How do they begin? Certain circumstances come together and lead to one action or another; or a thought comes to our mind to do something, and we do it. But the concurrence of circumstances does not come from us;  nor, obviously, is the thought to do something our own; somebody suggests it. Thus, in such cases, the origin or birth of the thought to do something cannot or should not be an object of self-praise. Yet how many actions are of this kind? If we examine them conscientiously, we shall find that they almost all start in this way. So we have nothing to boast of. (174)

This passage suggests the following argument: One cannot justifiably take credit for or take pride in anything (an action, a physical or mental attribute, etc.) unless one originates it "independently of anything else." But nothing is such that one originates it in sublime independence of all else. Therefore, one cannot justifiably take pride in anything. Here is a reason why pride is listed among the Seven Deadly Sins.

But does this amount to a metaphysical denial of the very existence of the self?  Not within Christian metaphysics. For that what is needed is the Buddhist assumption, crucial to the reasoning in the Anattalakkhana Sutta, that a self is an entity that has complete control over itself. Such a self could justifiably take pride in its actions and attributes. For it would be their fons et origo. So if one cannot justifiably take pride in any of one's actions or attributes, then one is not a self  in the sense in which this term is employed in the Anattalakkhana Sutta.  

In sum, both Buddha and Scupoli set the standard for selfhood very, and perhaps unattainably, high. Both claim that no one of us is a self for the reason than no one of us is in complete control of any of his actions or attributes. None of the things which one normally takes to be oneself or to belong to oneself (e.g., one's body, habits, brave decisions, brilliant insights, distinguished career, foolish mistakes, etc.) is such that one has originated it autonomously and independently.

Having set the standard for selfhood so high, Buddhism must deny that we are selves.  Christianity, however, is far less radical, holding as it does that we are selves all right, but ontologically derivative selves: we derive our being from the Creator, who is Being itself. (Deus est ipsum esse subsistens.) We are creatures of the one and only Absolute Self who brought us into being and sustains us in being. On Buddhism, nothing at all has 'self nature' in the eminent and plenary sense; on Christianity, one thing does, God. For Buddhism, there is no Absolute Self, there is in reality nothing like the Hindu Atman. For Christianity, there is an, or rather, the uniquely unique Absolute Self, namely, God. 

What about us? One thing is clear to both Buddhists and Christians: no one of us is identical to the Absolute Self.  For Buddhists, there is no Absolute Self, and for Christians, while there is an the Absolute Self, no one of us, no finite self, is or ever becomes identical to it, not even in the Beatific Vision.  If the visio beata involves a participation in the divine life, this does not involve a total assimilation: the finite self never loses its self-identity or individuality. In the Beatific Vision, creator-creature duality is mitigated, but never wholly eliminated.   

The main difference between Buddha and Scupoli is that the latter maintains that God gives us what we do not have under our control, our derivatively real selves. Thus for Scupoli, what we do not have from ourselves, we have from another, and so have. But for Buddha, what we do not have from ourselves, we do not have at all.

So in what sense does Scupoli embrace a 'no self' (anatta, anatman) doctrine? In the sense that in orthodox — miniscule 'o' — Christian metaphysics no one of us enjoys selfhood in the plenary sense of the term.