This is mainly covered in the Athrabeth Finrod ah Andreth (published in Morgoth's Ring) where the main text states:
Eru Himself must at some time come to oppose Melkor. But Eru could not enter wholly into the world and its history ... Finrod therefore thinks that He will, when He comes, have to be both 'outside' and inside; and so he glimpses the possibility of complexity or of distinctions in the nature of Eru, which nonetheless leaves Him 'The One'.
This is clearly based on the Christian concept of the Holy Trinity; in particular the Wikipedia article contains the following statement which is almost an exact paraphrase of Tolkien's last line above:
The three persons are distinct, yet are one "substance, essence or nature".
Regarding the Secret Fire, or Flame Imperishable, the passage I've quoted contains a direction to Author's Note 11, which I'll quote in full:
This is actually already glimpsed in the Ainulindale', in which reference is made to the 'Flame Imperishable'. This appears to mean the Creative activity of Eru (in some sense distinct from or within Him), by which things could be given a 'real' and independent (though derivative and created) existence. The Flame Imperishable is sent out from Eru, to dwell in the heart of the world, and the world then Is, on the same plane as the Ainur, and they can enter into it. But this is not, of course, the same as the re-entry of Eru to defeat Melkor. It refers rather to the mystery of 'authorship', by which the author, while remaining 'outside' and independent of his work, also 'indwells' in it, on its derivative plane, below that of his own being, as the source and guarantee of its being.
So the Flame Imperishable is seen as being part of, or indeed one with, Eru, but yet separate from him and representing his creative force which he put into the world and which he used to give life to his Children. The Ainulindale actually makes it explicit (thus making definite Tolkien's "appears to mean" above) that the Flame Imperishable is the means by which other things may be created, when discussing the origins of Melkor's Fall:
He had gone often alone into the void places seeking the Imperishable Flame; for desire grew hot within him to bring into Being things of his own...
So that already gives us a Trinity in Arda: Eru, the Flame Imperishable, and Finrod's prophecy of Eru entering into the world, which when compared to the Christian Trinity may correspond (in order) to the Father, the Spirit and the Son.
Getting back to the Flame Imperishable and it's use as a creative mechanism, the Nicene Creed (used in Catholic prayer) contains the following line:
We believe in the Holy Spirit, the Lord, the giver of life
Again, and especially with the "giver of life" part, this can be seen to have a correspondance.
There are other correspondances in the Athrabeth, relating to the words "breath", "fire" and "spirit", which may be viewed in the light of the Greek pneuma, used in the Bible texts, and translated spirit, wind or life, so Tolkien is definitely using similar terminology throughout this work.
However, this is all something that Tolkien himself was hugely uneasy about, and CT cites a note in his commentary on the Athrabeth:
Already it is (if inevitably) too like a parody of Christianity.
Although these are all from works not published until after Kilby's book, it is known that Kilby worked with Tolkien in 1964, that he had access to the Silmarillion papers (he was indeed the first "outsider" to see it), that the Athrabeth was in existence by that time (CT dates it to 1959), and that Tolkien viewed it as being part of the Silmarillion; as CT notes:
On one of these wrappers my father added: 'Should be last item in an appendix' (i.e. to The Silmarillion).
It is therefore likely that Kilby saw and read the Athrabeth.
I'm not aware of any place where Tolkien explicitly stated that the Flame Imperishable actually was Arda's version of the Holy Spirit, so I read Kilby's words as saying that Tolkien just described it in similar terms, although I don't have his book and so can't comment on what his exact statement may have been.
2014-09-08: Added in reference to this question: Was there a trinity in Eru?
Since I dealt with the Holy Spirit above, it's reasonable to ask if the Trinity is completed with the addition of a Son in Tolkien's writings. The answer is "yes", again sourced from the Athrabeth:
'What then was this hope, if you know?' Finrod asked.
'They say,' answered Andreth: 'they say that the One will himself enter into Arda, and heal Men and all the Marring from the beginning to the end. This they say also, or they feign, is a rumour that has come down through years uncounted, even from the days of our undoing.'
This does serve to reinforce Tolkien's "parody of Christianity" opinion, and may explain why CT omitted the Athrabeth from the published Silmarillion, despite it being, as he notes in his introduction:
... a major and finished work, and is referred to elsewhere as if it had for my father some 'authority'...