I checked the Lithuanian White pages on-line at www.takas.lt
for surnames beginning with KASS. I just found one listing and he's Russian.
When I first saw the double-S spelling I thought it might not be the original spelling.
I then tried KASAKAIT. There were 17 people listed with the Kasakaitis surname including married females (spelled Kasakaitiene) and unmarried females (spelled Kasakaityte).
All Lithuanian married female surnames end in IENE. Unmarried female surnames can end in AITE, YTE, or UTE depending upon the family surname spelling. Men's surnames remain the same regardless of marital status.
There are likely more people with that surname since the White Pages only has listed land lines and many Lithuanians just use their mobile phone numbers.
The surname distribution seemed to cluster around Marijampole in southwestern Lithuania. It's an area where my mother's parents came from.
While they may have left Russia (occupied Lithuania) illegally, they would have had Russian identity papers which would have shown their surnames and given names. I have my grandfather's Russian internal passport from circa 1907 which he carried with him when he came to the U.S.
Kassakitas is likely an Anglicized spelling which made it non-Lithuanians to spell.
Hope this helps in your search. Would love for you to find relatives in Lithuania.
Merry Christmas (Linksmu Kaledu - Lithuanian version).