According to many sources, the answer would be no. John Bunyan's theological beliefs were Puritan, which differed from pure Calvinism.
The concept of a covenant or contract between God and his elect pervaded Puritan theology and social relationships. In religious terms, several types of covenants were central to Puritan thought.
The Covenant of Works held that God promised Adam and his progeny eternal life if they obeyed moral law. After Adam broke this covenant,
God made a new Covenant of Grace with Abraham (Genesis 18-19).
Covenant of Grace. This covenant requires an active faith, and, as such, it softens the doctrine of predestination (Emphasis mine as this addresses the question directly). Although God still
chooses the elect, the relationship becomes one of contract in which
punishment for sins is a judicially proper response to disobedience.
During the Great Awakening, Jonathan Edwards later repudiated Covenant
Theology to get back to orthodox Calvinism. Those bound by the
covenant considered themselves to be charged with a mission from God.
Covenant of Redemption. The Covenant of Redemption was assumed to be preexistent to the Covenant of Grace. It held that Christ, who
freely chose to sacrifice himself for fallen man, bound God to accept
him as man's representative. Having accepted this pact, God is then
committed to carrying out the Covenant of Grace. According to Perry
Miller, as one contemporary source put it, "God covenanted with Christ
that if he would pay the full price for the redemption of beleievers,
they should be discharged. Christ hath paid the price, God must be
unjust, or else hee must set thee free from all iniquities" (New
England Mind 406).
All of this is evident in the Pilgrim's Progress. While I agree the two questions are different, the theology in Pilgrim's Progress is inseparable from the author, meaning he wrote as he believed. Puritan theology does not teach irresistible grade, in the purest sense. In Puritan thought, we clearly have a choice to respond to God's call. He calls us first, but we can choose to respond or not. This is more inline with the Armenian view.
I also managed to find a quote fro him grappling with the subject of predestination here: http://pohkok.blogspot.com/2009/01/predestination-troubles-john-bunyan.html Since it's not copyright, I'll copy the quote in it's entirety. Something I would not normally do, but since this is a link to a blog, and the likelihood of a future broken link is higher with blogs...
In his book "Grace Abounding to the Chief of Sinners", John Bunyan grappled with predestination:
By these two temptations I was very much afflicted and disquieted;
sometimes by one, and sometimes by the other of them. And first, to
speak of that about my questioning my election, I found at this time,
that though I was in a flame to find the way to heaven and glory, and
though nothing could beat me off from this, yet this question did so
offend and discourage me, that I was, especially sometimes, as if the
very strength of my body also had been taken away by the force and
power thereof. This scripture did also seem to me to trample upon all
my desires; It is not of him that willeth, nor of him that runneth;
but of God that showeth mercy (Romans 9:16).
With this scripture I could not tell what to do: for I evidently saw,
unless that the great God, of His infinite grace and bounty, had
voluntarily chosen me to be a vessel of mercy, though I should desire,
and long, and labour until my heart did break, no good could come of
it. Therefore this would stick with me, How can you tell that you are
elected? And what if you should not? How then?
O Lord, thought I, what if I should not indeed? It may be you are not,
said the Tempter; it may be so indeed, thought I. Why then, said
Satan, you had as good leave off, and strive no farther; for if
indeed, you should not be elected and chosen of God, there is no talk
of your being saved; For it is not of him that willeth, nor of him
that runneth; but of God that showeth mercy.
By these things I was driven to my wits' end, not knowing what to say,
or how to answer these temptations: (indeed, I little thought that
Satan had thus assaulted me, but that rather it was my own prudence
thus to start the question): for that the elect only attained eternal
life; that, I without scruple did heartily close withal; but that
myself was one of them, there lay the question.
Thus therefore, for several days, I was greatly assaulted and
perplexed, and was often, when I have been walking, ready to sink
where I went, with faintness in my mind; but one day, after I had been
so many weeks oppressed and cast down therewith as I was now quite
giving up the ghost of all my hopes of ever attaining life, that
sentence fell with weight upon my spirit, Look at the generations of
old, and see; did ever any trust in God, and were confounded?
At which I was greatly lightened, and encouraged in my soul; for thus,
at that very instant, it was expounded to me: Begin at the beginning
of Genesis, and read to the end of the Revelations, and see if you can
find, that there were ever any that trusted in the Lord, and were
confounded. So coming home, I presently went to my Bible, to see if I
could find that saying, not doubting but to find it presently; for it
was so fresh, and with such strength and comfort on my spirit, that it
was as if it talked with me.
Well, I looked, but I found it not; only it abode upon me: Then did I
ask first this good man, and then another, if they knew where it was,
but they knew no such place. At this I wondered, that such a sentence
should so suddenly, and with such comfort and strength, seize, and
abide upon my heart; and yet that none could find it (for I doubted
not but that it was in holy scripture).
Thus I continued above a year, and could not find the place; but at
last, casting my eye upon the Apocrypha books, I found it in
Ecclesiasticus 2:10. This, at the first, did somewhat daunt me; but
because by this time I had got more experience of the love and
kindness of God, it troubled me the less, especially when I considered
that though it was not in those texts that we call holy and canonical;
yet forasmuch as this sentence was the sum and substance of many of
the promises, it was my duty to take the comfort of it; and I bless
God for that word, for it was of God to me: that word doth still at
times shine before my face.
After this, that other doubt did come with strength upon me, But how
if the day of grace should be past and gone? How if you have overstood
the time of mercy? Now I remember that one day, as I was walking in
the country, I was much in the thoughts of this, But how if the day of
grace is past? And to aggravate my trouble, the Tempter presented to
my mind those good people of Bedford, and suggested thus unto me, that
these being converted already, they were all that God would save in
those parts; and that I came too late, for these had got the blessing
before I came.
Now I was in great distress, thinking in very deed that this might
well be so; wherefore I went up and down, bemoaning my sad condition;
counting myself far worse than a thousand fools for standing off thus
long, and spending so many years in sin as I had done; still crying
out, Oh! that I had turned sooner! Oh! that I had turned seven years
ago! It made me also angry with myself, to think that I should have no
more wit, but to trifle away my time, till my soul and heaven were
Note: For those who might be interested, this is Ecclesiasticus 2:10
in KJV: Look at the generations of old, and see; did ever any trust in
the Lord, and was confounded? or did any abide in his fear, and was
forsaken? or whom did he ever despise, that called upon him?
However, this is not without dispute. A differing opinion can be found here: http://www.founders.org/journal/fj82/article2.html The author here makes an argument for the case that Bunyan was, indeed, a five-point Calvinist.
John Bunyan believed that the Scriptures teach that God's intention in
the atonement was the redemption of the elect and them alone, and that
this was fully and effectually accomplished on the cross. This
conviction regarding the intention and accomplishment of the atonement
is evident throughout his writings, but it becomes most clearly and
maturely articulated in his later works, particularly as he
reflects upon the active obedience of Christ, the high priesthood of
Christ, and covenant theology.
Justification by faith alone is the heart of the gospel and the
Christian life for John Bunyan. He defended it on numerous occasions
from Ranter and Quaker errors, and this doctrine finds expression, in
some form or fashion, in almost every tract or treatise he published.
For Bunyan, Christ's vicarious obedience not only applied to His
death, but His life as well. Christ not only bore the sins of the
elect; He fulfilled the whole law in their stead as well. Thus in
his later works one can find Bunyan's commitment to limited atonement
clearly articulated in various descriptions of Christ's vicarious
obedience or active obedience on behalf of the elect.
Thus, the answer is "we're not sure" and it depends highly on which sources you choose to believe. IMO, the evidence points to, as defined in the Puritan Beliefs section that he didn't completely reject the doctrine of predestination, but rather had a "softened view" on it, meaning that he likely believed in the more Armenian understanding of resistible grace.