**Solution:**

To find the coordinates of vertex D, we need to understand the properties of a square.

**Properties of a square:**

1. All sides of a square are equal in length.
2. Opposite sides of a square are parallel.
3. Opposite angles of a square are equal and each angle measures 90 degrees.
4. Diagonals of a square bisect each other at right angles.

Using these properties, we can find the coordinates of vertex D.

**Step 1: Find the length of the sides of the square**

The distance between points A and B gives us the length of one side of the square.

Distance between A(-2, 3) and B(-2, -4):

Side AB = √((-2 - (-2))^2 + (3 - (-4))^2) = √(0^2 + 7^2) = √49 = 7

So, the length of one side of the square is 7 units.

**Step 2: Determine the direction of the square**

Since the opposite sides of a square are parallel, we can determine the direction of the square based on the slope of AB.

Slope of AB = (y2 - y1)/(x2 - x1) = (-4 - 3)/(-2 -(-2)) = -7/0

The slope is undefined, which means the line AB is vertical. Therefore, the square is oriented vertically.

**Step 3: Use the length and direction to find vertex D**

Since the square is oriented vertically, the x-coordinate of vertex D will be the same as the x-coordinate of point C(5, -4).

So, the x-coordinate of vertex D is 5.

To find the y-coordinate of vertex D, we can add the length of one side to the y-coordinate of point C.

y-coordinate of vertex D = y-coordinate of point C + length of one side

y-coordinate of vertex D = -4 + 7 = 3

Therefore, the coordinates of vertex D are (5, 3).

Hence, the correct answer is option D) (5, 3).