Born in Banja Luka, but spent his childhood in Kraljevo, he started out at local club Čibukovac. He also spent some time at OFK Beograd, before joining the youth system of Partizan. In the summer of 2015, Bošnjak was promoted to the first-team squad in preparations for the 2015–16 season.[1] He was subsequently loaned to their affiliated side Teleoptik in early 2016.
In November 2016, Bošnjak signed his first professional contract with Partizan, on a three-year contract.[2] He was again sent on loan to Teleoptik in the 2017 winter transfer window, helping them win the Serbian League Belgrade and promotion to the Serbian First League.
In the meantime, he played one season each in Voždovac and Kolubara before he returned to his hometown and signed for Borac Banja Luka in September 2021.