GCD of several integers
There are some issues, such as not using obvious built-in solutions, an inefficient algorithm and what looks like a logic error possibly caused by having odd array-growing logic.

I'll start with the error. In int min = numbers[1]; you assume that there are at least two numbers. That may not be true, if the user immediately enters 0 - which raises the question of what the GCD of the empty set is (0 or 1, depending on who you believe), but whatever it should be, it's bad form to crash with an out-of-bounds exception.

The odd array-growing logic is probably part of that problem, it leaves a spurious 0 in the array which you thereafter actively ignore - it's simpler to not have it there to begin with. Anyway, you wanted an array that can change size - that's usually called ArrayList which, conveniently, already exists. You can also avoid saving up the input entirely, by computing the GCD while reading the input.

The GCD algorithm is strange and inefficient. It's not the familiar GCD algorithm, which is already a strike against it by itself since that makes it hard to tell what it's doing, and there is no redeeming quality. This is an algorithm with a very bad worst case, as much as O(min*N) (if the GCD turns out to be 1). I strongly recommend using the modulo-based version of the Euclidean algorithm, which is efficient and recognizable, or maybe Binary GCD if you want something different. In the unlikely case that you rejected it because it takes only two inputs, of course gcd(a, b, c) = gcd(gcd(a, b), c) and so on, so you can simply loop over your input and call GCD on the gcd-so-far and the number from the input.

Or, as mentioned, you could compute the GCD while reading the input, something like this:

int gcdSoFar = 0;

while (true) {
    int number = input.nextInt();
    if (number == 0)
        break;

    gcdSoFar = gcd(number, gcdSoFar);
}

System.out.println("The GCD of the input is: " + gcdSoFar);


This loses the print-back of the input, if you want to preserve that you could use an ArrayList. By the way, in Java 8 you can use String.join(", ", list) to make a comma-separated string from your list, you don't have to do that manually.