Swift electrons passing near or through metallic structures have proven to be an excellent tool for studying plasmons and other types of confined optical modes involving collective charge oscillations in the material structures hybridized with electromagnetic fields. In this work, we provide a general analytical framework for the simulation of electron energy-loss spectroscopy (EELS) in infinite systems with cylindrical symmetry, such as wires, holes, and optical fibers. While EELS theory is well developed for electrons moving parallel to the direction of translational symmetry, we introduce closed-form analytical solutions for perpendicular electron trajectories. These analytical results are corroborated by comparison to numerical simulations based on a frequency-domain boundary-element method and a discontinuous-Galerkin time-domain finite-element method. Numerical solutions further allow us to study termination effects in finite-sized cylindrical objects such as nanorods. The present study of the interaction between free electrons and cylindrically symmetric photonic systems can find application in the analysis of EELS spectra and the design of free-electron–photonic hybrid systems.