Beta-solenoids are a class of protein nanotube structures that are observed in virulence factors, prion proteins and amyloid fibrils. Here we investigate the compressive strength of the triple-beta-helix solenoid structure found in the cell puncture needle of the bacteriophage T4 virus. We characterize the compressive mechanical strength of this protein nanotube using full-atomistic molecular dynamics simulations in explicit solvent over a wide range of deformation speeds. We observe that the dynamical behavior, stiffness and failure strength of the structure are strongly dependent on the deformation rate. We illustrate that H-bond rupture initiation is the atomistic mechanism that leads to instability and buckling of the protein nanotube at the peak force. We show that the behavior of the protein under small compressive deformation can be approximated by a rate-dependent linear elastic modulus, which can be used in context of a continuum Euler buckling formula for the triple-helix geometry to predict the failure load. Our work provides a link between the structure and biofunctional properties of this beta-solenoid topology, and illustrates a rigorous framework for bridging the gap between experimental and simulation time-scales for future compression studies on proteins. Our study is relevant to self-assembling peptide nanotube materials, and may provide insight into the influence of mechanical properties on the pathological pathways of virulence factors, prions and amyloids found in neurodegenerative diseases.