  We consider the periodic defocusing cubic nonlinear Klein-Gordon equation in
three dimensions in the symplectic phase space $H^{\frac{1}{2}}(\mathbb{T}^3)
\times H^{-\frac{1}{2}}(\mathbb{T}^3)$. This space is at the critical
regularity for this equation, and in this setting there is no global
well-posedness nor any uniform control on the local time of existence for
arbitrary initial data. We prove a local-in-time non-squeezing result and a
conditional global-in-time result which states that uniform bounds on the
Strichartz norms of solutions imply global-in-time non-squeezing. As a
consequence of the conditional result, we conclude non-squeezing for certain
subsets of the phase space, and in particular, we prove small data
non-squeezing for long times. The proofs rely on several approximation results
for the flow, which we obtain using a combination of probabilistic and
deterministic techniques.
