Epitaxial spin-filter tunnel junctions based on the ferromagnetic semiconductor europium monoxide (EuO) are investigated by means of density functional theory. In particular, we focus on the spin transport properties of Cu(100)/EuO(100)/Cu(100) junctions. The dependence of the transmission coefficient and the current-voltage curves on the interface spacing and EuO thickness is explained in terms of the EuO density of states and the complex band structure. Furthermore, we also discuss the relation between the spin transport properties and the Cu-EuO interface geometry. The level alignment of the junction is sensitively affected by the interface spacing, since this determines the charge transfer between EuO and the Cu electrodes. Our calculations indicate that EuO epitaxially grown on Cu can act as a perfect spin filter, with a spin polarization of the current close to 100%, and with both the Eu-5d conduction-band and the Eu-4f valence-band states contributing to the coherent transport. For epitaxial EuO on Cu, a symmetry filtering is observed, with the Δ1 states dominating the transmission. This leads to a transport gap larger than the fundamental EuO band gap. Importantly, the high spin polarization of the current is preserved up to large bias voltages.