The context of this passage, and the chapter as a whole, is about St Paul reaffirming the centrality of the Resurrection for the Christian faith. In verse 15:17, St Paul famously says that if Christ was not Resurrected then Christianity is a farce, and thus we're still dead in our sins. The JWs like to zero in on verses 44-50 since it is here that St Paul makes various statements that could be taken to deny the bodily Resurrection in favor of a purely spirit-creature resurrection. For example, St Paul says, “It is sown a physical body, it is raised up a spiritual body. If there is a physical body, there is also a spiritual one. It is even so written: “The first man Adam became a living soul.” The last Adam became a life-giving spirit. Nevertheless, the first is, not that which is spiritual, but that which is physical, afterward that which is spiritual. … I say, brothers, that flesh and blood cannot inherit God’s kingdom, neither does corruption inherit incorruption.” (15:44-46, 50 NWT)
Now for someone reading this for the first time, this certainly does come off as denying a physical Resurrection. Even those Christians who can sense that this isn't denying the bodily Resurrection but instead is speaking of Jesus' new state still might not know how to explain this or defend against a JW. The key to understanding this text is to keep in mind that St Paul is speaking of not just a human body, but a glorified human body. Grace 'elevates' (or 'builds on') human nature; it doesn't supplant human nature. So when it is said that Jesus Resurrected in His original body, that is true, but His Resurrected body was graced with the gift of incorruption and special powers, which is why the nailmarks in his hand and side did not bleed when He showed the Apostles. So when St Paul speaks of Christ as having a “spiritual” body, all this means is a glorified human body. The JW mind cannot see this crucial distinction, and only see it as an ordinary bringing back to life versus a bringing back as an angelic creature.
To prove this from the text, consider the verses that come next:
51 Behold! I tell you a mystery. We shall not all sleep, but we shall all be changed, 52 in a moment, in the twinkling of an eye, at the last trumpet. For the trumpet will sound, and the dead will be raised imperishable, and we shall be changed. 53 For this perishable body must put on the imperishable, and this mortal body must put on immortality. 54 When the perishable puts on the imperishable, and the mortal puts on immortality, then shall come to pass the saying that is written: “Death is swallowed up in victory.”
The Greek verb for "put on" here means putting on an article of clothing. This term need not refer exclusively to material clothing, as St Paul frequently uses this same term in reference to "putting on Christ" (Rom 13:14; Gal 3:27) and "putting on the armor of God" (Rom 13:12; Eph 6:11). So what we see here in verses 53-54 is an explanation of what is going to happen: perishable "puts on" imperishable and mortal "puts on" immortality. Since Protestants deny the Nature-Grace-Distinction, these texts will sound kind of weird, but not to Catholic ears! This means that when Paul was speaking of a spiritual body earlier, he was speaking of a body that had "put on" ("clothed itself" with) immortality. This leads to a final point. When the JWs read "flesh and blood cannot inherit God's kingdom" (v50), they read this as saying a physical body cannot possibly enter the spiritual realm of Heaven, so the physical body must be cast off. Of course, this interpretation is very incorrect, and the correct interpretation is plain: the phrase "flesh and blood" refers to the "mortality" that Paul says must "put on immortality"!
Knowing this information is essential to being able to not only explain the truth to a JW, but also to protect family members and friends who might find such claims (initially) appealing.