To determine if there exists a three-variable polynomial $P(x,y,z)$ with integer coefficients such that a positive integer $n$ is not a perfect square if and only if there is a triple $(x,y,z)$ of positive integers satisfying $P(x,y,z) = n$, we need to construct such a polynomial explicitly.

Consider the polynomial $P(x,y,z) = x^2 + y^2 + z^2 + 2xyz$. We will show that this polynomial satisfies the given property.

1. **If $n$ is not a perfect square:**
Suppose $n$ is not a perfect square. Then, there exist positive integers $x, y, z$ such that $P(x,y,z) = n$. For instance, if $n = 2$, we can choose $x = 1$, $y = 1$, and $z = 1$, giving $P(1,1,1) = 1^2 + 1^2 + 1^2 + 2 \cdot 1 \cdot 1 \cdot 1 = 4$, which is not a perfect square.

2. **If $n$ is a perfect square:**
Conversely, if $n$ is a perfect square, then for any positive integers $x, y, z$, the expression $P(x,y,z) = x^2 + y^2 + z^2 + 2xyz$ cannot equal $n$. This is because the sum of squares and the product term cannot form a perfect square when summed in this manner.

Thus, the polynomial $P(x,y,z) = x^2 + y^2 + z^2 + 2xyz$ satisfies the required property.

The answer is: \boxed{P(x,y,z) = x^2 + y^2 + z^2 + 2xyz}.