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The quantum capacity of bosonic Gaussian quantum channels can be non-additive in a particularly striking way: a pair of such optical fiber type channels can individually have zero quantum capacity but super-activate each other such that the combined channel has strictly positive capacity. This has been shown in Smith et al (2011 Nature Photon. 5 624) where it was conjectured that squeezing is a necessary resource for this phenomenon. We provide a proof of this conjecture by showing that for gauge covariant channels a Choi matrix with positive partial transpose implies that the channel is entanglement-breaking. In addition, we construct an example which shows that this implication fails to hold for Gaussian channels which arise from passive interactions with a squeezed environment.