Wordorigins.org

This self-referential American term is of uncertain origin. The most likely source of the word is the Dutch nickname Janke, a diminutive of Jan (John).  Although it must be said that this is not the only possible explanation.

Use as a nickname for a person is attested to from the 1680s. The Calendar of State Papers, Colonial Series, an archive of British government documents, has this from 1683:

They sailed from Bonaco..; chief commanders, Vanhorn, Laurens, and Yankey Duch.