Summary
Bumfuck, Egypt is first documented in army slang from 1972, and BFE is from at least 1988. Bumfuck, [Egypt] appears to be the original, followed by variations Bumblefuck (1989), Bubblefuck (1993) and Buttfuck (1999).
Sometimes these variations appear standing alone, sometimes in Egypt, or Africa (also found in BFA), or another country, (rural) state or place, and sometimes prefixed by East or West. More recently, both BFN and Butt Fucking Nowhere are reported from 2002.
As Bumfuck, Egypt appears to be originally military slang, given the number of abbreviations they used I can easily believe it was also jocularly and euphemistically shortened to BFA before 1988.
Partridge
The Concise New Partridge Dictionary of Slang and Uncoventional English (2003):
BFE noun any remote location. An abbreviation of BUMFUCK, EGYPT US 1989
Bumblefuck noun any remote, small town US 1989
Bumfuck, Egypt noun a mythical town that is the epitome of remoteness. With variants US 1972
It also gives BFN and Butt Fucking Nowhere, both from 2002.
OUP
These 1989 and 1972 first citations are given with others in The F-Word (2009, Oxford University Press):
BFE noun [Bumfuck (or Bumbfuck), Egypt]
Military & Students. a very remote place; the middle of nowhere.
1989 P. Munro U.C.L.A. Slang 20: Troy...lives out in B.F.E....Bum Fuck, Egypt.
Bumfuck noun
Military & Students. a very remote place. --used with a placename, esp. in Bumfuck, Egypt. Also Bumfuckistan. Jocular. See also BFE, BUMBLEFUCK.
1972 Sgt. E-6, U.S. Army: They probably sent those records out to Bumfuck, Egypt.
It also gives Bumblefuck as a 1989 alteration of Bumfuck; Bubblefuck as a 1993 alteration of Bumblefuck; and Buttfuck, usually used with a placename, from 1999.
Usenet
I found an earlier example for the initials BFE in Usenet, on 11th September 1988 in alt.cyberpunk:
$600 is about what we paid for my first computer -- a C64, 1541 and printer; when I was a upper-lower class kid living in the backwoods of
BFE, Leesville, Louisiana. (Ft. Polk sux too. :-). We weren't on
welfare, but we weren't rich either. I think we saved about 9-12months
for the computer. (Grandparents bought me a monitor a month later. What
a rescue. :-). That computer made the difference between me being
a dual major CompSci/Journalism student in Houston instead of being
like my cousins: 2-4 kids, HS diploma at best, slow paying job, no future,
and living in BFE, Louisiana.
According to Wordorigins.org the theory is as follows...
1960s student slang according to etymology online. Some sites suggest that it’s origin might be from being so drunk that your face is in or near a toilet. I don’t think that can be assumed as fact, however.
Not sure how true it is, as such expressions are seldom tracked.
Best Answer
In rural America, which was an agrarian society originally, people did not have indoor plumbing and instead used an outhouse. They didn't have toilet paper either, so they had to use whatever was at hand. Corncobs were often used for the purpose. That's how they came to be associated with the rectum.
EDIT: What I said above was related to me by my grandfather, who grew up in a farming community in Illinois in the early 20th century. So it may be folk etymology. Etymonline.com differs somewhat, although I find their explanation bizarre (and note that it is qualified with the word "apparently" — apparently what?
FURTHER EDIT:
Etymonline offers no explanation for how "anal intercourse" became associated with an innocuous game. Was there a gay culture on the down-low in farming communities, one that staged mock games of what amounts to bean-bag tossing to cover up homosexual orgies? That seems ridiculous.