Poquette is a 1970 graduate of Memorial High School in his hometown of Eau Claire, Wisconsin. Playing on the varsity team in three sports, he was a starter in both baseball and football for three years and basketball for two. He was a member of the Wisconsin Interscholastic Athletic Association baseball champions during his sophomore year in 1968. He was eventually honored as one of 25 charter inductees into the school's athletics hall of fame in 2005. He was also among the second induction class into the Eau Claire Baseball Hall of Fame in 2009.

He was selected in the fourth round (80th overall) by the Royals in the 1970 MLB draft. He was a reserve on the Royals' American League (AL) West title teams in 1976, 1977 and 1978 and appeared in the League Championship Series in each of those three years.