Měli byste zabalit whereBetween &orWhereBetween in where. Zkuste toto:
$childs = Program::whereIn('id', $programs_by_date)
->where('shareable', '=','1')
->where(function($query) use ($new_start_date, $new_end_date){
$query->whereBetween('starting_date', [$new_start_date,$new_end_date])
->orWhereBetween('ending_date', [$new_start_date,$new_end_date]);
})
->orderBy('starting_date')
->get();
Doufám, že to bude fungovat.