Born in Mercedes, Eugui began playing football with Bristol de Mercedes. Soon, he joined Club Nacional de Football, where he won the 1969 Uruguayan Primera División, and C.A. Cerro in 1970. Eugui had a brief stint in Argentina with Argentinos Juniors before moving to Mexico in 1972, where he scored a goal in the semi-finals of the 1972 CONCACAF Champions' Cup and won the league with Deportivo Toluca F.C. in 1974–75.
After retiring from playing, Eugui became a football manager. During his time at Ciudad Juarez, Eugui helped ht team not to be relegated to Primera A, and the team advanced to the Clausura 2009 semifinals after they eliminated defending champions Club Toluca, but they were eliminated by CF Pachuca by an aggregate score of 4-3. For the next tournament after 10 games and only just tied 4 games and lost 6 without a win he was sacked. He managed Ciudad Juarez in 44 games, won 12, tied 12 and lost 15.
On May 17, 2011, Eugui was named the new coach of Toluca for the Apertura 2011 season. In 17 games, he managed to win 4, tie 8, and lose 5 games. After the season ended, Eugui was sacked from the team.
On 13 November 2017, he was named the manager of Potros UAEM for the Clausura 2018 season.
