One way to get the squares, as Victor asked in a comment, is the following: a simple module over $\mathfrak{sl}_2\oplus\mathfrak{sl}_2$ is of the form $V_n\otimes V_m$ (where $V_n$ is the $\mathfrak{sl}_2$-module of dimension $n+1$), and this has a submodule (for $\mathfrak{sl}_2$ acting diagonally) of dimension one exactly when $n=m$.
One way to get the squares, as Victor asked in a comment, is the following: a simple module over $\mathfrak{sl}_2\oplus\mathfrak{sl}_2$ is of the form $V_n\otimes V_m$ (where $V_n$ is the $\mathfrak{sl}_2$-module of dimension $n+1$), and this has a submodule of dimension one exactly when $n=m$.