  Universal features in the scalings of Shannon-R\'enyi entropies of many-body
groundstates are studied for interacting spin-$\frac{1}{2}$ systems across
(2+1) dimensional $O(3)$ critical points, using quantum Monte Carlo simulations
on dimerized and plaquettized Heisenberg models on the square lattice.
Considering both full systems and line shaped subsystems, $SU(2)$ symmetry
breaking on the N\'eel ordered side of the transition is characterized by the
presence of a logarithmic term in the scaling of Shannon-R\'enyi entropies,
which is absent in the disordered gapped phase. Such a difference in the
scalings allows to capture the quantum critical point using Shannon-R\'enyi
entropies for line shaped subsystems of length $L$ embedded in $L\times L$
tori, as the smaller subsystem entropies are numerically accessible to much
higher precision than for the full system. Most interestingly, at the quantum
phase transition an additive subleading constant $b_\infty^{*\rm line}=0.41(1)$
emerges in the critical scaling of the line Shannon-R\'enyi entropy
$S_\infty^\text{line}$. This number appears to be universal for 3d $O(3)$
criticality, as confirmed for the finite-temperature transition in the 3d
antiferromagnetic spin-$\frac{1}{2}$ Heisenberg model. Additionally, the phases
and phase transition can be detected in several features of the participation
spectrum, consisting of the diagonal elements of the reduced density matrix of
the line subsystem. In particular the N\'eel ordering transition can be simply
understood in the $\{S^z\}$ basis by a confinement mechanism of ferromagnetic
domain walls.
