Gene isn't exactly a model protagonist; he is at times his own worst enemy. His fear, resentment, and jealousy get the best of him time and time again. When he fails to return Finny's declaration of friendship on the beach, when he plots against Finny academically, when he wants his friend to get in trouble, and when he jounces the limb, Gene reveals these antagonistic qualities, these weaknesses of character.
Brinker isn't exactly an antagonist, but his repeated accusations against Gene mean he's making trouble for our protagonist. It's not absolutely clear why Brinker acts as he does, whether out of a sense of justice, a curiosity about the truth, or (a less attractive option) a sadistic desire to make Gene suffer.