Kveinys was awarded the Soviet Master of Sport title for chess in 1981.

In 2015, Kveinys tied for first place in the U.S. Masters Chess Championship.

In 2016, he won the Chess Festival in Bad Wörishofen.

[6] He was awarded the FIDE titles of International Master in 1990 and Grandmaster in 1992.

He was a graduate of the chess department of the Moscow Institute of Physical Culture and Sports (now Russian State University of Physical Education, Sport, Youth and Tourism).