An overpowering right-handed relief pitcher, Boever played in Major League Baseball from 1985 to 1996. His MLB debut was with the St. Louis Cardinals on July 19, 1985,  as the Redbirds were closing in on the National League pennant. He contributed to
the Cardinals pennant winning drive during their stretch run in 1985. That Cardinals team advanced to the World Series. He pitched for the Cardinals in 1985 and 1986, the Atlanta Braves from 1987 to 1990, the Philadelphia Phillies from 1990 to 1991, the Houston Astros in 1992, the Oakland Athletics in 1993, the Detroit Tigers from 1993 to 1995, and the Pittsburgh Pirates in 1996.

In 1992, he led the league in appearances for a pitcher, playing in 81 games for the Houston Astros. Known for his devastating palm ball, he was a closer for the 1989 Atlanta Braves, collecting 21 saves. 

In 1985, Boever pitched for the Double A Arkansas Travelers and the Triple A Louisville
Cardinals before being called up to the Major Leagues with the St. Louis Cardinals.

In 1982, Boever pitched for the Springfield Cardinals and the Erie Cardinals. He was with the St. Petersburg Cardinals in 1983 and 1984. He wrapped up his playing career in 1997 with the Calgary Cannons and the Edmonton Trappers in the Pacific Coast League.

In the late 1970s, Boever pitched for Lindbergh High School, and he was inducted into the Lindbergh Athletics Hall of Fame in 1993. Before signing with the Cardinals in 1982, Boever pitched for Meramec Community College, Crowder College, and the University of Nevada at Las Vegas.