Božić began his professional career with the Croatian first level side NK Kamen Ingrad for whom he appeared 34 times in the 2005–06 and 2006–07 1.HNL seasons,[1] before moving abroad and joining the Bosnian side NK Široki Brijeg.

In August 2009, he returned to Croatia and joined the Druga HNL side HNK Suhopolje on a free transfer.

[2] He then joined the Prva HNL side HNK Cibalia Vinkovci where he was a regular first-team member for one season.

After an unsuccessful trial with Viktoria Plzeň,[3] he signed with Dukla Prague, a team newly promoted to the Czech Gambrinus liga, in January 2012.

[4] Božić scored his first Gambrinus liga goal for Dukla in March 2013, in the second half of a 4–0 home win against the bottom-of-the-league team České Budějovice.