  We study the scaling of the (basis dependent) Shannon entropy for
two-dimensional quantum antiferromagnets with N\'eel long-range order. We use a
massless free-field description of the gapless spin wave modes and phase space
arguments to treat the fact that the finite-size ground state is rotationally
symmetric, while there are degenerate physical ground states which break the
symmetry. Our results show that the Shannon entropy (and its R\'enyi
generalizations) possesses some universal logarithmic term proportional to the
number $N_\text{NG}$ of Nambu-Goldstone modes. In the case of a torus, we show
that $S_{n>1} \simeq {\rm const.} N+ \frac{N_\text{NG}}{4}\frac{n}{n-1} \ln{N}$
and $S_1 \simeq {\rm const.} N - \frac{N_\text{NG}}{4} \ln{N}$, where $N$ is
the total number of sites and $n$ the R\'enyi index. The result for $n>1$ is in
reasonable agreement with the quantum Monte Carlo results of Luitz et al.
[Phys. Rev. Lett. 112, 057203 (2014)], and qualitatively similar to those
obtained previously for the entanglement entropy. The Shannon entropy of a line
subsystem (embedded in the two-dimensional system) is also considered. Finally,
we present some density-matrix renormalization group (DMRG) calculations for a
spin$\frac{1}{2}$ XY model on the square lattice in a cylinder geometry. These
numerical data confirm our findings for logarithmic terms in the $n=\infty$
R\'enyi entropy (also called $-\ln{p_{\rm max}}$). They also reveal some
universal dependence on the cylinder aspect ratio, in good agreement with the
fact that, in that case, $p_{\rm max}$ is related to a non-compact free-boson
partition function in dimension 1+1.
