Yes they did, though the majority who did escape were not Chinese but either Uyghurs or Mongols who fled East Turkestan and Inner Mongolia respectively. In particular, Uyghurs either fled due to religious reasons, as the CCP began to close down mosques, or for political or social reasons seeing that the majority who fled were in some ways the "top of the society" or were inordinately going to be punished for some slight. Many of the Kazakhs who had initially fled the Soviet union, also wound up going back during the Cultural Revolution.

Sources:

• Eurasian Crossroads: A History of Xinjiang by James A. Millward
• Islam in China: Hui and Uyghurs Between Modernization and Sinicization by Jean A. Berlie
• China, Xinjiang, and Central Asia: History, Tradition, and Crossborder Interaction Into the 21st Century by Colin Mackerras and Michael Clarke
• Xinjiang - China's Muslim Far Northwest by Michael Dillon
• A Sino-Soviet Cultural Frontier: The Ili Kazakh Autonomous Chou by George Moseley