Is solving the following system of boolean equations NP-hard?
[Now that the question's been clarified I'll post my previous comment as an answer.]
It's in $\mathsf{P}$. Start with unit propagation. Afterwards, what's left on the right-hand sides will be monotone, so will be satisfied by setting all remaining variables to True.
(If you want to count the number of solutions, on the other hand, that might be harder.)