Swindell attended the University of Texas at Austin where he was a member of Sigma Nu Fraternity. According to the Texas Longhorns media guide, he was one of the most decorated pitchers in school history, with a 43-8 record in 77 games and a 1.92 ERA. He made 50 starts, pitching 32 complete games and notching school records for shutouts (14) and strikeouts (501). He remains among the top 10 all-time in Longhorns history for ERA (10th), victories (3rd), innings (2nd), strikeouts (1st), appearances (4th), starts (3rd), complete games (3rd) and shutouts (1st). Swindell also had 13 career saves, which ranks sixth at UT and the top two single-season strikeout totals in UT history (204 in 1985 and 180 in 1986). He helped UT capture three straight Southwest Conference titles, post at least 51 wins during each of his three seasons, and finish second at the College World Series in 1984 and 1985.
