According to many sources, the answer would be no. John Bunyan's theological beliefs were Puritan, which differed from pure Calvinism.

From "Puritanism in New England" by Donna Campbell:

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 grace, 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 in line with the Arminian view.

I also managed to find a quote from him grappling with the subject of predestination in his memoir Grace Abounding to the Chief of Sinners (found in the blog post Predestination Troubles John Bunyan):

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 at some times, 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 sheweth mercy' (Rom. 9.16).

With this scripture I could not tell what to do; for I evidently saw, that unless 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 still 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 further; for if, indeed, you should not be elected and chosen of God, there is no talk of your being saved; 'For it is neither of him that willeth, nor of him that runneth, but of God that sheweth 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 all 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 the Lord, and was 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 was ever any that trusted in the Lord, and was 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 I was as if it talked with me.

Well, I looked, but I found it not; only it abode upon me; then I did 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 it was in Holy Scripture.

Thus I continued above a year, and could not find the place; but at last, casting my eye into 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 into the country, I was much in the thoughts of this, But how if the day of grace be 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 was I 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 lost.

However, this is not without dispute. A differing opinion can be found in the Founders Journal article "John Bunyan and the Extent of the Atonement." The author 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,5 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.5 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. In my opinion, the evidence points to Donna Campbell's statement 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 Arminian understanding of resistible grace.

It's clear from the writings of John Bunyan that he believed in the Reformed doctrines of predestination and irresistible grace. He makes an extensive defense of these and related doctrines in his work, Come and Welcome to Jesus Christ, in which he analyzes John 6:37:

All that the Father giveth me shall come to me; and him that cometh to me I will in no wise cast out. [KJV]

Demonstrating his belief in predestination, he writes that the "all" of this verse are the elect:

Those, therefore, intended as the gift in the text, are those that are given by covenant to the Son; those that in other places are called “the elect,” “the chosen,” “the sheep,” and “the children of the promise,” &c. These be they that the Father hath given to Christ to keep them; those that Christ hath promised eternal life unto; those to whom he hath given his word, and that he will have with him in his kingdom to behold his glory. (5; 8)

He sees the "giving" of this verse as not only something happening in the present, with the transformation of individual sinners, but also something that God did long ago:

this gift was bestowed upon Christ when the covenant, the eternal covenant, was made between them before all worlds. (6; 11)

Turning now to the doctrine of irresistible grace, we see that Bunyan also clearly teaches it in this work. He spends many pages expounding on the "shall come" of John 6:37, arguing that it is an absolute promise of God that cannot be circumvented by anything, not even the will of man:

The Father’s end was, that they might come to him, and be saved by him; and that, says the Son, shall be done; neither sin nor Satan, neither flesh nor world, neither wisdom nor folly, shall hinder their coming to me. (9; 19)

But what of the one who insists that he will not be saved? Is it possible for him to resist God's will? Bunyan says no:

Here, then, is the case; we must now see who will be the liar, he that saith, "I will not"; or he that saith, "He shall come to me." "You shall come," says God; "I will not come," saith the sinner. Now, as sure as he is concerned in this Shall-come, God will make that man eat his own words; for "I will not," is the unadvised conclusion of a crazy-headed sinner; but Shall-come was spoken by him that is of power to perform his word. [bold and quotation marks added] (22; 51)

How will this be done? Will such people be dragged into the kingdom against their will? On the contrary, says Bunyan:

The obstinacy and plague that is in the will of that people, shall be taken away; and they shall be made willing; Shall-come will make them willing to come to thee. (22; 52)

Summary

John Bunyan's formulations of predestination and irresistible grace do not diverge from the historical teaching of the Reformed faith. It's true that he differed with Calvinists on some matters related to ecclesiology and the sacraments, but his understanding of salvation corresponds neatly with theirs.

Quotes come from Come and Welcome to Jesus Christ, a PDF published by bunyanministries.org. The first number after each quote indicates the page number in that edition. The second number after each quote refers to the page number in a 1774 edition of this work.