The 32-year-old Ribeiro had 18 goals and 45 assists in 74 games last season.
The veteran of 12 NHL campaigns will be joining his third NHL team. He spent six seasons with the Montreal Canadiens and another six with the Stars.
Ribeiro will be paid $5 million next season in the final year of a five-year, $25 million contract.
Eakin played in 30 games last season in his rookie campaign, with four goals and four assists. He also played 43 games in the AHL with the Hershey Bears.
Eakin was Washington's third round pick in the 2009 NHL Draft. He has two years remaining on his entry-level contract.