Using the slow-cooling method in selected fluxes, we have grown spontaneously nucleated single-crystals of pure GeO2 and SiO2-substituted GeO2 materials with the alpha-quartz structure. These piezoelectric materials were obtained in millimeter size as well-faceted, visually colorless, and transparent crystals. Cubic-like or hexagonal prism-like morphology was identified depending on the chemical composition of the single-crystals and on the nature of the flux. Both the silicon substitution rate and the homogeneity of its distribution were estimated by Energy Dispersive X-ray spectroscopy. The cell parameters of the flux-grown GeO2 and Ge1-xSixO2 (0.038 <= x <= 0.089) solid-solution were deduced from their X-ray powder diffraction pattern. As expected, the cell volumes decrease as the silicon content substitution increases. A room temperature Infrared spectroscopy study confirms the absence of hydroxyl groups in the as-grown crystals. Unlike what was observed for hydrothermally grown GeO2 crystals, these flux-grown oxide materials did not present any phase transition before melting as pointed out by a Differential Scanning Calorimetry study. Neither a alpha-quartz/beta-quartz transition as encountered in SiO2 near 573 degrees C nor a alpha-quartz to rutile transformation were detected for these GeO2 and Ge1-xSixO2 single-crystals.