As a Chinese name:
The Jung 鄭 surname comes from Chinese people of the Yin-Ping region of Guang-dong province in China.
Jung is the Cantonese pronunciation of the Chinese family name 鄭 for those immigrants coming to North America in the 50-60s. It
More recently it is also sometimes written as Chang or Cheng in the Hong Kong dialect. In Mandarin the name is often written as Zheng.
As a Germanic name:
The Jung surname is also a common Germanic name that is pronounced as "Young" rather than with the English "J" sound.