No. Basal Cell Carcinoma (BCC) and Squamous Cell Carcinoma (SCC) do not have the potential to become melanoma. BCC, SCC, and Melanoma are different forms of skin cancer that originate from distinct and separate skin cell types. BCC originates from the basal cells in the skin (epidermis), which get their name from being at the base of the epidermis. SCC originates from cells called keratinocytes within the epidermis. When either the basal cells or the keratinocytes become cancerous, they can invade deeper levels of the skin. However, BCC tends not to spread to other parts of the body. SCC of the skin is more likely than BCC to spread (metastasize), but fortunately this is also infrequent. On the other hand, melanoma originates from cells called melanocytes. Melanocytes are the skin cells that make melanin and give the skin its color. Unlike BCC and SCC, Melanoma has a high tendency to become metastatic and spread to other parts of the body.
Each cell type in the skin has a potential to "go bad" and become cancerous, but one cell type does not turn into another cell type. Therefore, BCC and SCC do not have the potential to evolve into melanoma. However, it is possible for one individual to have more than one type of skin cancer on different parts of the body.