The answer to this question is more complicated than it might appear. Eddie Eagan, as the question states, won gold medals for boxing at the Summer Olympics in 1912 and for bobsleigh at the Winter Olympics in 1924. But Sweden's Gillis Grafström won gold medals at the Summer Olympics in 1920 and the Winter Olympics in 1924 and 1928. The difference is that Grafström's medals were all in the same event – namely figure skating, which was contested at the 1920 Summer Olympics but in the Winter Olympics thereafter.
Grafström is the only competitor to win individual medals at both the Summer and Winter Olympics, since Eagan's Winter Olympic gold was in the four–man bobsleigh.
© Haydn Thompson 2017