From 1999 to 2005, Eyre made his way through the Twins' minor league system, playing for the rookie-level Elizabethton Twins, Single-A Quad City River Bandits, High-A Fort Myers Miracle, Double-A New Britain Rock Cats, Phoenix Desert Dogs (Arizona Fall League), and Triple-A Rochester Red Wings.

Eyre spent all of the 2010 season with the Triple-A Oklahoma City RedHawks, accumulating a 5-4 record and 3.50 ERA with 59 strikeouts across 49 relief appearances.

[3] He made 39 appearances for the Triple-A Sacramento River Cats in 2011, registering a 4-5 record and 3.48 ERA with 45 strikeouts and 9 saves across 62 innings of work.

[10][11][12] Eyre was named after Willie Mays by his mother, who was an admirer of the Hall of Fame center fielder despite being a Los Angeles Dodgers fan.

He also has a younger brother, Robert Grace, who is playing in the minor leagues for the San Francisco Giants organization.