Will the white patches (hypopigmentation) that follow an exzema outbreak go away?

Yes: Usually hypopigmentation from inflammation such as in eczema will resolve once the eczema is treated, but it can take time. Remember, that area of skin may tan differently than the surrounding skin. Wear your sunscreen and sun protection to minimize that contrast and avoid burns. Note: if scars from deep scratching exists, those color changes may be permanent.