I first heard this circa 1970, long before Al Gore invented the internet (speaking of myths). It’s obvious that neither NASA nor anyone else has a time machine, and that ordinary retrocalculation could not yield any such result.
It has been around for 125 years, at least:
http://www.truthorfiction.com/joshuaday/
[snip] It is interesting to note that an attempt to explain Joshua’s long day from a scientific standpoint was published many years before Harold Hill’s NASA story. In the 1930s, Dr Harry Rimmer wrote a book called ‘The Harmony of Science and Scripture’, in which he drew from an 1890 book by Yale professor C.A. Totten.
Using popular biblical chronologies of the time, Totten concluded that the world was created 4,000 years before the birth of Christ, on Sunday, September 22, 4000BC. However, he said that the calendar calculations showed that September 22 was actually a Monday, and not a Sunday, and that the error was probably because of Joshua’s missing day.
Totten also makes reference to the Hezekiah story in 2 Kings and argues that this accounts for another missing 20 minutes or so. Totten’s calculations sound suspiciously familiar to the NASA story. The bottom line seems to be that there is a lot still missing about ‘the Missing Day’. [/snip]