Beagle was born in Hartford, Connecticut but played high school football in Cincinnati. The 6 foot one inch, 185 pounder entered the United States Naval Academy in 1952. A physical player, Beagle won All-America recognition in his junior and senior seasons during an era in which ends played 60 minutes. He was honored with the Maxwell Award after his junior year in 1954, in which he had 30 receptions, 451 yards rushing and four touchdowns to compliment his stout defense. He finished his three-year career with 64 receptions for 849 yards (mostly from George Welsh) and eight touchdowns despite fighting through a broken hand.
Beagle married a Navy nurse, Jo Ann Jones. Their daughter, Ronda J. Beagle, graduated from the Naval Academy in 1984. The Beagles were among the first families with a father and daughter who graduated from the Academy. Ronda Beagle authored a 1993 book on the Navy's efforts to recruit women.
Uniquely in the history of the sport, Beagle and Roger Staubach played football for the same high school (Purcell in Cincinnati), played college ball at the same school (Navy), won the Maxwell Trophy, made All-America, and eventually were inducted into the College Football Hall of Fame, with Beagle receiving that honor in 1986.