We experimentally verify the quantum non-Gaussian character of a conditionally generated noisy squeezed single-photon state with a positive Wigner function. Employing an optimized witness based on probabilities of squeezed vacuum and squeezed single-photon states, we prove that the state cannot be expressed as a mixture of Gaussian states. In our experiment, the non-Gaussian state is generated by conditional subtraction of a single photon from a squeezed vacuum state. The state is probed with a homodyne detector and the witness is determined by averaging a suitable pattern function over the measured homodyne data. Our experimental results are in good agreement with a theoretical fit obtained from a simple yet realistic model of the experimental setup.