While these aren't the same as your question, the following threads may be interesting:

https://www.reddit.com/r/AskHistorians/comments/1o7qlt/when_did_the_rural_us_alabama_georgia_texas_etc/

https://www.reddit.com/r/AskHistorians/comments/3a9406/when_did_the_modern_version_of_the_redneck/

So to start with Redneck/Hillbilly are troublesome terms. Sometimes it's used as a term simply for an individual who labors out in the sun and other times it's a slur for poor, uneducated people from a rural background and everything in between.

Second American "Hillbilly" culture isn't really a monolothic entity, the culture of someone in the mountains of WV will be different from that of someone from the swamps of LA. Though the two would be more similar to each other than they would to that of someone who grew up in Brooklyn (which in turn is closer to say that of someone from Compton than LA while still being distinctly different).

But despite that there is a belief and identification with a more national Hillbilly/Redneck identity and I would identify this as really becoming established at the tail end of the 1960's as a backlash to the West-coast centered counterculture. Though most elements of the culture were certainly around prior to this and I think /u/aDoge's response hits on this. This reaction can I think best be understood by examining some music of the time so I'm going to pick and choose some songs as examples, to represent the attitudes that were being expressed and clashing.

So on the West Coast (and across the pond in England) we had all these bands singing about sex, drugs, rock'n'roll. And kids are going out, getting naked, communes are a thing, it's a crazy time where all these more traditional values are being rejected. And then you've got songs like "Eve of Destruction"

Think of all the hate there is in Red China

Then take a look around to Selma, Alabama

You may leave here for four days in space

But when you return it's the same old place

The pounding of the drums, the pride and disgrace

You can bury your dead but don't leave a trace

Hate your next door neighbor but don't forget to say grace

Add to this events like the Altamont Free Concert (which admittedly happened at the very tail end of the 60's) where the Rolling Stones had the Hells Angels providing security. During the concert the Stones sang their song Sympathy for the Devil and a massive fight broke out, then a concert goer drew a gun, and a Hell's Angel stabbed him to death (he was later acquitted of murder charges).

So we've got this scary (and potentially violent) West Coast Counter-Culture that seemed to just spring out of nowhere saying "it's the fault of people like YOU" that the world's gone to pot to this large group of people who were primarily rural/southern and religious and trying to corrupt the youth. And remember all this is happening in the midst of the civil rights movement and many of these smaller communities that we might identify as Hillbilly towns were on the side of segregation and Jim Crow so they're already feeling under attack.

So they're feeling like they're under siege from every angle and when they get the opportunity to rally together around a common identity against the counter-culture they take it. The, in my mind, quintessential example of this song is Merle Haggards Okie from Muskogee. Lyrics. I recomend you look over the lyrics, it's literally a point by point rejection of every aspect of the counter-culture, basically a counter-counterculture. And this phenomenon wasn't restricted to music, these exact same people were the ones being appealed to by Nixon with his speech about the "silent majority". Nixon was reassuring people that they were not in fact on the Eve of Destruction, that all these hippies were a small but vocal majority (and if they came out and voted for him he'd set things right again).

So that's one answer to what Hillbilly/Redneck culture is, though I need to emphasize it's not the only one and will change depending on how the term is defined. I have known "rednecks" whose personal beliefs are far more in line with those of hippies from the 1960's than the attitude expressed in "Okie from Muskogee" and vice versa. But the idea of a national "redneck" identity seems to have grown out of a reactionary backlash to the counterculture of the 1960's and ballooned from there.

Sorry for the lack of sources and rambling, rushed to get this down on my lunch break , if the mods don't delete it I'll try to add sources in 7 hours or so when I get home.

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