I've seen the McCormick envelopes applied many times to the product of two continuous variables, but I can't seem to find when both of them are binaries. Also, I applied the restrictions as described bellow, and they don't work because they don't make sense when both of them are 1.
the restrictions: $$ \begin{align*} w_{i j} &\geq x_{i}^{L} \cdot x_{j} + x_{i} \cdot x_{j}^{L} - x_{i}^{L} \cdot x_{j}^{L}\\ w_{i j} &\geq x_{i}^{U} \cdot x_{j} + x_{i} \cdot x_{j}^{U} - x_{i}^{U} \cdot x_{j}^{U}\\ w_{i j} &\leq x_{i}^{U} \cdot x_{j} + x_{i} \cdot x_{j}^{L} - x_{i}^{U} \cdot x_{j}^{L}\\ w_{i j} &\leq x_{i} \cdot x_{j}^{U} + x_{i}^{L} \cdot x_{j} - x_{i}^{L} \cdot x_{j}^{U}\\ x^{L} &\leq x \leq x^{U} \qquad~ w^{L} \leq w \leq w^{U} \end{align*} $$
Substituting $0$ for $x^L$ and $1$ for $x^U$ yields: \begin{align} w_{i j} &\ge 0 \\ w_{i j} &\ge x_j + x_i - 1 \\ w_{i j} &\le x_j \\ w_{i j} &\le x_i \end{align} This is the usual linearization of $w_{ij} = x_i x_j$. See https://or.stackexchange.com/questions/37/how-to-linearize-the-product-of-two-binary-variables