I just use whatever shampoo I have. Brushes are made out of hair, or if synthetic, the shampoo won't hurt them. Shampoo is designed to get rid of oil & grease and product.
Brushes usually fall apart if there is residual water left in the ferrule (the metal part.) The glue holding the bristles in place may get soggy and the bristles fall out. Always shake the excess water out of the bristles in a downward motion to get the water out of there. Then shape the bristles to dry.