This is an ancient custom which predates Christianity. Pliny (Natural History 28.5) records a custom among the Romans of greeting someone who sneezes:
Why is it that we salute a person when he sneezes, an observance which Tiberius Caesar, they say, the most unsociable of men, as we all know, used to exact, when riding in his chariot even? Some there are, too, who think it a point religiously to be observed to mention the name as well of the person whom they salute. [...] These usages have been established by persons who entertained a belief that the gods are ever present, in all our affairs and at all hours, and who have therefore found the means of appeasing them by our vices even.1
So he says that superstitions of this kind are attributable to the belief that certain apparently meaningless events are actually "signs" from some spiritual power. Homer also gives an account of a sneeze-omen in the Odyssey 17.528-550, where Penelope predicts that Odysseus will kill her suitors if he returns: at this point, Telemachus sneezes (Τηλέμαχος δὲ μέγ' ἔπταρεν), and Penelope interprets that as a sign that her prediction is true.
The fact that sneezing is involuntary might make it more likely for people to see a hidden hand at work. Augustine (On Doctrine 2.20) writes about "the most frivolous practices" of superstitious observances when "any part of the body should jump", or "to go back to bed if any one should sneeze when you are putting on your slippers". 2 In this case, the sneeze is interpreted as a divine warning that today is a bad day to do anything important.
Because a sneeze comes from the head, the seat of reason, it may be regarded as being of divine origin, according to the school of Aristotle; and moreover, it was seen as a sign of good health and vitality, unlike coughing, which was more commonly associated with disease (Problems chapter 33, "Concerning the nose"). The only Biblical sneeze is in 2 Kings 4, where Elisha restores a dead boy to life, demonstrating a perceived link with the breath of life:
When Elisha came into the house, he saw the child lying dead on his bed. So he went in and closed the door on the two of them, and prayed to the Lord. Then he got up on the bed and lay upon the child, putting his mouth upon his mouth, his eyes upon his eyes, and his hands upon his hands; and while he lay bent over him, the flesh of the child became warm. He got down, walked once to and fro in the room, then got up again and bent over him; the child sneezed seven times, and the child opened his eyes. 2 Kings 4:32-35 (NRSV)
Just from these examples, we can see that sneeze customs are a bit ambiguous when it comes to the sneeze being "good" or "bad". "God bless you!" could be an appropriate response either way. But Scripture also often speaks against the interpretation of omens - or at least encourages people to seek actual divine guidance - so we could say that a "get well soon" kind of "bless you" is acceptable, but the "ooh, that's unlucky" kind is not.
I have heard the evil nasal demon explanation many times but I have never seen a source which endorsed the theory. (It's not hard to find people sneering at the silly mediaevals, but this may be yet another case where they weren't so silly.)
1. Original: cur sternuentes salutamus? quod etiam Tiberium Caesarem, tristissimum, ut constat, hominum, in vehiculo exegisse tradunt, et aliqui nomine quoque consalutare religiosius putant? [...] haec instituere illi, qui omnibus negotiis horisque interesse credebant deos et ideo placatos etiam vitiis nostris reliquerunt. English translation above by John Bostock and H. T. Riley (Taylor and Francis, 1855).
2. Original phrases: inanissimarum observationum; si membrum aliquod salierit; redire ad lectum si quis dum se calceat sternutaverit.