If a complex number $z = x + yi$ is Gaussian integer, then $|z|^2 = x^2 + y^2$ is the sum of two squares of integers.
The square of an even integer is divisible by 4, while the square of an odd integer gives remainder 1 when divided by 4. Therefore, the sum of squares of two integers can give remainder 0, 1 or 2 when divided by 4.
If $n$ is greater than 3, then among any four consecutive positive integers in the sequence
$$|z_1|^2, |z_2|^2, \dots, |z_n|^2$$
there would be a number giving the remainder 3 when divided by 4, which is impossible. Hence, $n$ is less than or equal to 3.
Notice that $2 + 2i$, $3$ and $3 + i$ are Gaussian integers whose absolute values squared are $8$, $9$ and $10$, respectively. Therefore, the answer for $n$ is $3$.