The diagram below shows an isosceles triangle with base $21$ and height $28$.  Inscribed in the triangle is a square.  Find the area of the shaded region inside the triangle and outside of the square.

[asy]
size(170);
defaultpen(linewidth(0.8));
draw((0,0)--(1,1));
pair A=(5,0),B=(-5,0),C=(0,14), invis[]={(1,2),(-1,2)};
pair intsquare[]={extension(origin,invis[0],A,C),extension(origin,invis[1],B,C)};
path triangle=A--B--C--cycle,square=(intsquare[0]--intsquare[1]--(intsquare[1].x,0)--(intsquare[0].x,0)--cycle);
fill(triangle,gray);
unfill(square);
draw(triangle^^square);
[/asy]
1. **Identify the dimensions of the isosceles triangle:**
   - The base of the triangle is \(21\) units.
   - The height of the triangle is \(28\) units.

2. **Determine the area of the entire triangle:**
   \[
   \text{Area of the triangle} = \frac{1}{2} \times \text{base} \times \text{height} = \frac{1}{2} \times 21 \times 28 = 294 \text{ square units}
   \]

3. **Set up the problem for the inscribed square:**
   - Let \(s\) be the side length of the inscribed square.
   - The square is inscribed such that its top side is parallel to the base of the triangle and its bottom side lies on the base of the triangle.

4. **Use similar triangles to find the side length \(s\):**
   - The triangle can be divided into two smaller triangles and a rectangle (the square).
   - The height of the smaller triangle above the square is \(28 - s\).
   - The base of the smaller triangle above the square is \(\frac{21 - s}{2}\).

5. **Set up the ratio using similar triangles:**
   - The ratio of the height to the base of the smaller triangle is the same as the ratio of the height to the base of the entire triangle.
   \[
   \frac{28 - s}{\frac{21 - s}{2}} = \frac{28}{21/2}
   \]
   Simplify the right-hand side:
   \[
   \frac{28}{10.5} = \frac{56}{21} = \frac{8}{3}
   \]

6. **Solve for \(s\):**
   \[
   \frac{28 - s}{\frac{21 - s}{2}} = \frac{8}{3}
   \]
   Cross-multiply to solve for \(s\):
   \[
   3(28 - s) = 8 \left(\frac{21 - s}{2}\right)
   \]
   \[
   3(28 - s) = 4(21 - s)
   \]
   \[
   84 - 3s = 84 - 4s
   \]
   \[
   s = 12
   \]

7. **Calculate the area of the square:**
   \[
   \text{Area of the square} = s^2 = 12^2 = 144 \text{ square units}
   \]

8. **Calculate the area of the shaded region:**
   \[
   \text{Area of the shaded region} = \text{Area of the triangle} - \text{Area of the square} = 294 - 144 = 150 \text{ square units}
   \]

The final answer is \(\boxed{150}\)