As you may know, prior to the 20th century in the "Western World," not only the borders of countries but also people's surnames were much more fluid than they generally are today. Borders changed much more readily and many people were rather casual about their names. And spelling in particular was much less regimented. In addition, even today we see some differences in the way names are Anglicized. Remember when Beijing used to be called Peking?
Lithuania has variously been part of (or considered to be part of) either Poland or Russia in the last several centuries, and depending on who was in charge of a particular area, Lithuanians often changed their name to be more Polish or Russian.
The Lithuanian spelling of your family's name was probably Zinkavičius or Žinkevičius. Between the wars, that name was most prevalent in Kėdáiniai, in Kaunas county.
Zinkevicius is not an uncommon name today. Here's a guy on Facebook who's even from Kėdáiniai. https://www.facebook.com/viliu.zinkevicius