Pilar cyst or epidermoid cyst: These are common benign cysts that can present as a lump under the skin and may express a thick, black, or cheesy material when squeezed. The location on the breast and the description of the black secretion are consistent with this diagnosis.

Sebaceous cyst: Similar to pilar or epidermoid cysts, sebaceous cysts can also present with a blackhead appearance and express a thick, black secretion when squeezed.
Folliculitis or infected hair follicle: Although less likely given the description of only black secretion, an infected hair follicle could potentially present with a similar appearance, especially if the infection is deep and involves the sebaceous gland.
Keratosis pilaris: This condition causes small, rough patches on the skin and can sometimes be mistaken for blackheads, but it's less likely given the specific location and the nature of the secretion.

Breast cancer: Although extremely rare in a 12-year-old, any breast lump or unusual discharge warrants consideration of malignancy to ensure prompt diagnosis and treatment if present.
Inflammatory breast cancer: This rare and aggressive form of breast cancer can cause changes in the skin of the breast, including thickening or a pitted appearance, but it would be exceptionally rare in this age group.
Tuberculosis of the breast: This is a rare condition that can cause lumps, abscesses, or fistulas in the breast tissue and should be considered in endemic areas or in patients with risk factors.

Steatocystoma multiplex: A rare condition characterized by multiple sebaceous cysts, which could potentially present with similar symptoms.
Nevus comedonicus: A rare congenital condition that can cause a group of open comedones (blackheads) in a linear or other patterned distribution, but it's typically present from birth or early childhood.
Dermoid cyst: A type of teratoma that can occur on the breast and may contain sebaceous material, but it's less common and the presentation might differ.