Most stars form in clusters where relatively close encounters with other stars are common. Such encounters can leave architectural imprints on the orbits of planets and planetesimals residing in the interacting systems. This paper investigates the dependence of these features on the orbital parameters that describe the interaction. We explore the parameter space of stellar encounters with the outer Solar System using a large ensemble of N-body simulations. The results show that stellar encounters can leave unique signatures in the inclination distribution of a debris disk (the analog of the Kuiper Belt). We also derive analytical expressions that estimate the degree of inclination excitation in the hierarchical limit and thus describe the distribution of expected inclination excitation. In addition to their applications to general planetary systems, these results provide a possible explanation for the origin of the highly inclined (extreme) TNOs found in the outer Solar System. Moreover, the generally low inclinations observed in the Kuiper belt provide constraints on the number and severity of possible stellar flybys, and provide corresponding constraints on the properties of the birth cluster of our Solar System. These constraints can be made sharper in the future as observational surveys (e.g., DES, LSST) continue to discover new bodies in the outer Solar System.