Any $n \ge 5$ has the desired property: one can choose $a_1 = 1 - m$, $a_2 = a_3 = \dots = a_{n-3} = 0$, $a_{n-2} = -1$, $a_{n-1} = m$, $a_n = 0$.

Numbers $n = 3$ and $n = 4$ do not have the property.

For $n = 3$, $-2m = 2a_1a_2 + 2a_2a_3 + 2a_3a_1 = (a_1 + a_2 + a_3)^2 - a_1^2 - a_2^2 - a_3^2$ comes to $a_1^2 + a_2^2 + a_3^2 = 2m$, which does not hold for $m = 14$. Indeed, $28$ can not be written as a sum of three perfect squares.

For $n = 4$, $-m = a_1a_2 + a_2a_3 + a_3a_4 + a_4a_1 = (a_1+a_3)(a_2+a_4) = -(a_1+a_3)^2$ can hold only if $m$ is a perfect square, i.e. it does not hold for all $m \ge 0$.