If $z_1[n]$ is the response to input $x_1[n]$ (and $y[n]$ is NOT an input signal), and $z_2[n]$ is the response to $x_2[n]$, then it is straightforward to show that the response to a linear combination $ax_1[n]+bx_2[n]$ is NOT equal to $az_1[n]+bz_2[n]$, which would be a requirement for linearity. And due to symmetry, the same is the case if $y[n]$ is the input (and $x[n]$ is NOT an input signal). The test for time-invariance also fails in both of these cases, because only the input signal is shifted (and not the other nuisance signal).
When both $x[n]$ and $y[n]$ are input signals, the linearity test above still fails, so the system is non-linear (because it is an affine mapping due to the constants in the system equation). It is, however, time-invariant because if both input signals are shifted by some constant, the output is shifted by the same constant (which is not the case if only one of the two signals is an input signal).