Does hillbilly slang fall under a type of English language and if not, what is it called?

The answer to the question "is it a type of English" is definitely "yes". While the lines between "language" and "dialect" and "local slang" can blur, it's obvious that certain geographical areas have their particular style of speech, and these local variants are certainly studied as regional and local dialects, and not (as some people might claim) as "mistakes" or errors in speech.

Here's a good podcast episode by linguist John McWhorter about what constitutes a "dialect" vs a "language", if you want to learn more about what makes a "type of English".

And as for your specific dialect in question, "hillbilly" usually refers to residents of the Appalachian and Ozark mountains in the eastern USA, and they have a rich and quite distinct dialect with its distinct phonology and vocabulary. The Wikipedia entry for "Appalachian English" lists as references many books published on the topic, if you want to read up more on it.

I've voted up the "Appalachian English" answer, but depending on what you mean, there has historically been a larger sub-dialect area in the US referred to as "South Midland" with related or sub-dialects that include Appalachia and the Ozark highlands in Missouri^*. It has often been ethnically associated with the in-migration of the Scotts-Irish in America. There's a discussion of it on Robert Delaney's website. It can effectively be considered a "highlands" accent, which sounds more like standard Midlands the further from the rural hills you go.

A lot of South Midland isn't super noticeably different from the rest of Midland (which many consider "Standard American English"), except in a few details. Using "you all" for second person plural is my favorite, but linguists seem to prefer to talk about things like "cot-caught merger". However, in the Ozarks and Appalachia the difference is much more pronounced.

Second person plural map: (Its a bit tough to see, but there's a lot of yellow "You all" in that whiteish in-between area)

Cot-Caught merger map: Green is complete, yellow is in process.

Pen-pin merger map: At first glance this looks "southern", but then you notice it extends well up into standard Midlands areas like Kansas City and Indianapolis as well.

For background, I grew up in Tulsa, OK, and have been often told I have "no accent". However, for me "caught" and "cot" are homonyms, as are "pin" and "pen", and the proper second person plural in English is "you all". These are just facts, and I cannot be convinced otherwise.

^{* - Current linguistic opinion is that this dialect has slowly become subsumed by "North" Midland, and they are today similar enough to not be considered separate dialects. However, there's still lots of older recorded media out there (not to mention older human beings) from when that wasn't the case.}