To consolidate my comments: The proof of inequality (A.2) rests on the fact that the solutions $x^{k+1}$ and $z^{k+1}$ of the substeps (3.2) and (3.3), respectively, are global minimizers, so you can compare them to the saddle points $x^*$ and $z^*$, respectively. (Here, Boyd uses the subdifferential characterization, but I'm sure you can also derive it using the optimality directly.) But this is only guaranteed if the subproblems are convex (where any minimizer is a global minimizer), which is the case if $f$ and $g$ are convex.
That's not saying ADMM can't work for nonconvex problems -- indeed, this is currently a very hot topic in nonsmooth optimization and image processing -- but the fact that you can only work with local minimizers makes the analysis (and practice) significantly more involved.