In this talk, I will discuss symplectic non-squeezing for the nonlinear Klein-Gordon equation (NLKG) which can be (formally) regarded as an infinite dimensional Hamiltonian system. The symplectic phase space for this equation is at the critical regularity, 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 will present several non-squeezing results for the NLKG, including a conditional result which states that uniform bounds on the Strichartz norms of solutions for initial data in bounded subsets of the phase space imply global-in-time non-squeezing. The proofs rely on several approximation results for the flow, which we obtain using a combination of probabilistic and deterministic techniques.