Given Information:
The points A(3, 1), B(0, 4), C(-3, 1), D(0, -2) are vertices of a quadrilateral.

Approach:
To determine the type of quadrilateral formed by the given points, we can calculate the lengths of the sides and diagonals of the quadrilateral. Based on these calculations, we can identify the shape of the quadrilateral.

Calculating Side Lengths:
Let's calculate the lengths of the sides of the quadrilateral using the distance formula:
- Side AB: √[(x2 - x1)^2 + (y2 - y1)^2] = √[(0 - 3)^2 + (4 - 1)^2] = √[9 + 9] = √18
- Side BC: √[(x2 - x1)^2 + (y2 - y1)^2] = √[(-3 - 0)^2 + (1 - 4)^2] = √[9 + 9] = √18
- Side CD: √[(x2 - x1)^2 + (y2 - y1)^2] = √[(0 - (-3))^2 + (-2 - 1)^2] = √[9 + 9] = √18
- Side DA: √[(x2 - x1)^2 + (y2 - y1)^2] = √[(3 - 0)^2 + (1 - (-2))^2] = √[9 + 9] = √18

Calculating Diagonal Lengths:
Let's calculate the lengths of the diagonals of the quadrilateral using the distance formula:
- Diagonal AC: √[(x2 - x1)^2 + (y2 - y1)^2] = √[(-3 - 3)^2 + (1 - 1)^2] = √[36] = 6
- Diagonal BD: √[(x2 - x1)^2 + (y2 - y1)^2] = √[(0 - 0)^2 + (4 - (-2))^2] = √[36] = 6

Identifying the Quadrilateral:
- As we can see, all the side lengths of the quadrilateral are equal to √18, and both diagonals have equal lengths of 6.
- A quadrilateral with equal side lengths and equal diagonal lengths is a square.
- Therefore, the given quadrilateral with vertices A(3, 1), B(0, 4), C(-3, 1), D(0, -2) is a square.

Answer:
The given quadrilateral is a square (option B).