This is, of course, a great big load of codswallop.
"Napkin" is a Middle English word from French nappe, a tablecloth or towel, together with the English diminutive suffix -kin. It usually meant a table napkin. It appears in Tyndale's New Testament as napkyn, translating Greek soudarion. From Tyndale, it became the standard word in John 20:7 used by generations of translators, including by those of the KJV and RSV.
"Napkin" as an English word has also changed meaning since the days of William Tyndale. In American English, it still means primarily a table napkin, but in British English it usually means a diaper (a "nappie"). In Australian English a "napkin" is a feminine hygiene product (a table napkin is a "serviette").
Greek soudarion is a loanword from Latin sudarium, which meant a handkerchief or tablecloth. It was usually used of a bit of cloth used to wipe perspiration from the face. Not a table napkin, but then, in Tyndale's English, any squarish bit of linen used for any purpose could be called a napkin.
The use of it at Jesus' burial had nothing to do with dining. It was the custom then to bind the face around with a cloth to keep the jaw of the dead person from going slack. This custom was preserved in many cultures up to modern times. When the ghost of Jacob Marley encounters Ebenezer Scrooge, Scrooge "marked the very texture of the folded kerchief bound about its head and chin . . ." This face-cloth was folded or rolled and passed beneath the chin and then tied at the top of the head. It thus didn't go over the face, but around the face. In fact, the term used in John 20:7 is kephalé, "head," rather than prosópon, "face."
The fact that the soudarion was found by itself, apart from the shroud, would seem to refer to the manner of the resurrection of Jesus. Dorothy Sayers suggested that he didn't stand up, alive, and fling off the shroud, but simply disappeared from within it. It collapsed where it was, and the face-cloth rolled away by itself, since it had nothing to support it. That may or may not be how it happened. But it is quite clear that Jesus was wrapped in a shroud, which would normally have been bound about him with rolls of cloth. The face-cloth may have been bound about the head on the outside, or tied directly about the head on the inside; I don't know the custom. Since the other cloths are not mentioned, perhaps Jesus was not bound within his shroud, other than with the face-cloth; after all, the women were hoping to come later and complete his anointing for burial.
However that may be, the point is, this whole thing about a table napkin in the tomb and Jesus coming back to finish dinner is just sheer fantasy. It's what we call "eisegesis," meaning reading into the text a meaning that is not there. Eisegesis is opposed to the discipline of exegesis, meaning deriving the meaning of the text from what the text actually says. The idea that this table napkin nonsense is being pushed by Evangelical Christians makes it even worse. Evangelicals gripe at the Progressives for standing Scripture on its head, and then they come up with this stuff.
I know a lot of people -- preachers included -- who think learning Greek and studying history and all that is a distraction from sharing the gospel. But no matter how much you say you revere the Bible as the actual Word of God, if you approach it as if you are free to construe it however you can via the medium of your own colloquial (and frequently misunderstood) English, then you are making yourself the infallible authority, not the sacred book.
Making up pious little devotionals about Jesus being buried with a table napkin would be simply funny, if the stakes weren't so high. For if you can foist your meaning on the text, so can others. And there are a lot of people who find various teachings in the Bible to be out of step with their worldview, and who will tell you what the Bible "really" means. The only defense against false prophets is to be able to say what the Bible actually means.