As a variety of quantum computing models and platforms become available, methods for assessing and comparing the performance of these devices are of increasing interest and importance. Despite being built of the same fundamental computational unit, radically different approaches have emerged for characterizing the performance of qubits in gate-based and quantum annealing computers, limiting and complicating consistent cross-platform comparisons. To fill this gap, this work proposes a single-qubit protocol (Q-RBPN) for measuring some basic performance characteristics of individual qubits in both models of quantum computation. The proposed protocol scales to large quantum computers with thousands of qubits and provides insights into the distribution of qubit properties within a particular hardware device and across families of devices. The efficacy of the Q-RBPN protocol is demonstrated through the analysis of more than 300 gate-based qubits spanning eighteen machines and 2000 annealing-based qubits from one machine, revealing some unexpected differences in qubit performance. Overall, the proposed Q-RBPN protocol provides a new platform-agnostic tool for assessing the performance of a wide range of emerging quantum computing devices.