He played junior hockey with the Windsor Spitfires of the Ontario Hockey League and college hockey with the Michigan Wolverines. Eaves was drafted by the Winnipeg Jets in the 3rd round, 44th overall, in the 1980 NHL Entry Draft, Eaves spent most of his pro career in the minor pros, mostly in the American Hockey League. He spent 1990-94 playing for European teams and the Canadian National Team. He now coaches Bantam Tier 1 hockey and lacrosse at Shattuck - St. Mary's school in Faribault, Minnesota. Murray is married to Deborah J. Dybdahl.