Call the zeros of f(x) a and b, where f(x)=x^2-4x-7 (ignoring the absolute value).

When n is zero, the graph of f(x) is a concave upward parabola with zeros at a and b. It's minimum value is at (2, -11)

But the graph of |f(x)| has the part of the curve below the x-axis flipped or reflected across the x-axis into positive territory.  It has a relative maximum at (2,11).

When n=0, the equation (with or without the absolute value) is satisfied only at 2 points, a and b.

For n<0, the graph no longer intersects the x-axis and there are no solutions.  (the absolute value is never negative).

For n=11, the graph of f(x) would have 2 zeros, but the reflected part of |f(x)| is tangent to the x-axis at (2,0), so there are 3 solutions.

For n > 11 there are only the 2 solutions of the zeros of f(x).

If 1 <= n <= 10, then there are four real solutions.

The requested number of integer values for n is 10.

https://www.desmos.com/calculator/wulhrjxkcu