Can a healthy diet help prevent fibrocystic breast disease?

Fibrocystic disease: Yes and especially when all the nutrients are supplied. Avoid iodine deficiency in particular. You can do this by taking kelp tablets every day.

Normal breast tissue: All breasts are fibrocystic. When the breasts get sore and lumpy during your period, this is fibrocystic change. These changes can also occur due to increases in caffeine intake, stress and hormone use. So in essence, you can't get rid of fibrocystic breasts but you can improve the symptoms related to them (see some of my other posts on breast tenderness/fibrocystic breast disease).