My relatives in the the red clay country of central Georgia use this expression.

It does not mean "mind your own business". It means you, the speaker, are a disinterested party.

For example, say that two people are arguing a point about which you have an insight, or a fact to add. So as not to take sides you say: I don't have a dog the fight, but...blah, blah, blah.

So in this case, in my opinion, the female in question was mis-using the expression. She was likely indicating butt out, but I have never heard the expression used in that way by my folks in Georgia and South Carolina. Btw, my male cousins were avid coon hunters who owned packs of "coon dogs".