In this paper, we propose a Similarity-Quality-based Messenger Distillation (SQMD) framework for heterogeneous asynchronous on-device healthcare analytics. By introducing a preloaded reference dataset, SQMD enables all participant devices to distill knowledge from peers via messengers (i.e., the soft labels of the reference dataset generated by clients) without assuming the same model architecture. Furthermore, the messengers also carry important auxiliary information to calculate the similarity between clients and evaluate the quality of each client model, based on which the central server creates and maintains a dynamic collaboration graph (communication graph) to improve the personalization and reliability of SQMD under asynchronous conditions. Extensive experiments on three real-life datasets show that SQMD achieves superior performance.