Lennon deliberately chose to obfuscate the meaning of his signature song, even the title Strawberry Fields Forever was, in its day, controversial and ambiguous. Lennon simply took the name from a Salvation Army children's home in Liverpool called Strawberry Field (singular) which existed until 2005. The site has now been closed since 20011.
According to Wikipedia, Lennon's iconic lyric was inspired by a childhood memory
For the refrain, [...]: the words "nothing to get hung about" were inspired by Aunt Mimi's strict order not to play in the grounds of Strawberry Field, to which Lennon replied, "They can't hang you for it."
In other words, the young John Lennon knew full well he couldn't be arrested or given the death penalty for playing on the premises. The death penalty in the UK was suspended in 1965, one year after the last two men had been hanged in Britain, and five years after The Beatles had already been formed. The verb hang, when it refers to being killed by means of a rope tied to one's neck, is regular but it is not uncommon for native speakers (and in some dialects, I suppose) to use the irregular past tense hung in speech and in writing.
In fact, in verse two an even more obscure lyric, whose meaning never really caught on, is the following.
No one I think is in my tree, I mean it must be high or low
Does the tree refer back to hanging, or as John Lennon asserted in 1980 to his feelings of alienation, and “Therefore, I must be crazy or a genius — ‘I mean it must be high or low,’ ”
The origin of ‘get hung about’
As the OP and Xann have both stated, get hung about was not recorded in print before 1967.
However, the noun hang-up, also spelled hangup, meaning: a preoccupation, fixation, or psychological block; complex is first attested 1959, from notion of being suspended in one place. (Etymonline).
Cambridge Dictionaries lists hung up, and be hung up on sth: to be extremely interested in or worried by a particular subject and spend an unreasonably large amount of time thinking about it; Collins also has hung up, but mentions also hang about: to waste time; loiter, and the American expression, not care a hang about. Meanwhile, Oxford Dictionaries list the following entries under hang (US): not care (or give) a hang, and hang around. In the standard English dictionary, the second and third meanings of hung up about/on are supplied: Emotionally confused or disturbed, and Obsessed with or worried about
‘I don't get too hung up about business, I don't take it that seriously.’
Unfortunately, the online dictionary fails to mention its earliest appearance in print, so turn we must to Etymonline, which states
Hung-up is from 1878 as "delayed;" by 1961 as "obsessed."
The earliest instance of ‘to get hung about’ that I found on Google Books is dated 1969 [source], which was clearly inspired by the Beatles lyric
Here Wolfe discovers something to believe in. Something that is real, something to get hung about . . . .
The Beatles Bible.com: Strawberry Fields Forever
Wikipedia: Strawberry Field (the orphanage)