I don't pretend to know all of the facts, I am just speculating. But it is to say the least not very promising circumstances for successful home schooling. We are not talking about an educated mother fully able to formally teach her kids the three Rs. We are talking about a harried poor woman with a disabled husband who must spend her days tending goats, feeding chickens, and re-arranging the dismantled trash that constitutes her entire household wealth. Her kids may very well do nothing more for edcuation than help her with household chores and other such tasks.
It is indeed possible for a parent to neglect to educate her children. Whether that is the case here is an empirical question. Parents have duties. If they do indeed perform them, then sure the state should butt out. But if they fail to perform them, hiding behind a smokescreen of disagreement with state education to attract outside sympathy and support, that is not acceptable.
Those who support her side in all of this would be well advised to leave the crusading against evil socialism for next year, and to spend their efforts getting this family on its feet instead. It appears they barely have a house - a cast-off mobile home without proper sewage and utility hook-ups - or means of livelihood - they are herding goats, the husband collects disability. And they react to these straighten circumstances how? By picking fights with the state and filing lawsuits. That is real practical, isn't it?
None of which makes any of the other actors in this little drama into saints, needless to say...