Is NAE-HORN-SAT in P or NP-hard?
Your problem, NAE-HORN-SAT is $NP$-complete. Monotone NAE-SAT is $NP$-complete and it is a special case of your NAE-HORN-SAT. Literals in monotone SAT formula are either all positive or all negative.

Take any instance of Monotone NAE-SAT problem (all its literals are negative) then this formula is an instance of NAE-HORN-SAT problem.