Place the chicken breasts in the bottom of the slow cooker. Add the drained white beans, diced green chiles, diced onion, and minced garlic on top.
Pour in the chicken broth, then sprinkle the cumin, oregano, chili powder, salt, and black pepper over the mixture.
Cover and cook on low for 6 hours until the chicken reaches an internal temperature of 165°F and shreds easily.
Remove the chicken from the slow cooker and shred it with two forks. Return the shredded chicken to the pot.
Stir in the Greek yogurt until the chili becomes creamy and well combined. Let it sit for 5 minutes before serving.