The term "hyper-Calvinist" is a moving target, due to its pejorative nature, but several historical theologians are widely considered to fall under that label. I'll examine the views of two of them before moving to the position held by "traditional" Calvinists/reformed theologians. Both groups see God working to change the will of the sinner, making him desire repentance and salvation. But their understandings of who God calls to repentance differ.
Two men commonly considered hyper-Calvinists were John Brine and John Gill. They lived a century before Spurgeon, but they would have influenced Spurgeon's opponents. John Brine in particular objects to interpreting the call of Revelation 22:17 universally:
The Offer of Christ [...] is not made to Men, as Men, or to Sinners, merely as Sinners; but it is made to them as convinced, thirsting, hungring Sinners after Christ, and his Righteousness, and his Salvation.
The Elect [...] are the only Persons to whom [Christ] is held forth in the Gospel for Acceptance: Because they only have Desires after him.
Brine's concern is that if the offer of Christ is made to all men, it would impugn the sincerity of God:
If it be said, that such is the Will and Design of God, then it must be allowed, that he is disappointed of his Purpose in making this Proposal, with respect to the far greater Number of those to whom the Offer is made: Which seems to me evidently inconsistent with the Perfections of God. (source)
John Gill's analysis of the verse is similar:
the persons encouraged to partake of it are "whosoever will"; that is, whoever has a will to divine and spiritual things, wrought in him by God, for no man has such a will of himself (source)
Gill thus limits the invitation to those who have the will to do so, as given by God.
Among those commonly considered "traditional" Calvinists (i.e., not "hyper"), it is widely agreed that there is a "universal call," that is, the Gospel invitation is to be presented to everyone, not merely to those who will respond to it. However, some, like John MacArthur, still see Revelation 22's "whosoever will" as referring to those who have the God-given desire to come:
Whoever wants to, whoever desires to, whoever wishes to. You're thirsty and you wish to have your need met. That, by the way, is an unlimited invitation. Typical of the gracious and wide offer of salvation that you see in Scripture. It is essentially what Jesus said in John 6:37 where He said, "All that the Father gives Me shall come to Me and the one who comes to Me I will certainly not cast out." If you're thirsty it's because the Father has begun to move on your heart and you've recognized your need. And if you come because you want to take the water, it's because you've been prompted to come and there's no way the Lord would ever turn you back. So whoever wants to come, He receives. (source)
Others, like Albert Barnes, suggest that the invitation of this passage is truly to "all," even those who do not wish to be saved:
Every one that is disposed to come, that has any sincere wish to be saved, is assured that he may live. No matter how unworthy he is; no matter what his past life has been [...] the invitation is freely made to all to come and be saved. (source)
Charles Hodge cites this passage in his more general argument regarding the extent of the invitation:
The Scriptures, therefore, in the most explicit terms teach that the external call of the gospel is addressed to all men. [...] We are commanded to make the offer of salvation through Jesus to every human being on the face of the earth.
But what about Brine's objection? Hodge emphasizes the Scriptural evidence pointing to a universal call, and recognizes the incapacity of man to understand God's purposes. He then responds:
What God’s purposes may be in instituting and promulgating this scheme of mercy, has nothing to do with our duty as ministers in making the proclamation, or with our obligation and privilege as sinners in accepting his proffered grace. If it is not inconsistent with the sincerity of God to command all men to love Him, it is not inconsistent with his sincerity to command them to repent and believe the gospel. (source)
In other words, we may not understand why God invites all but saves only some, but Scripture and experience prove this to be true. And if it is legitimate for God to command all to obey him, but not make them all obedient, then it is certainly legitimate for God to command all to repent but not make them repent.