In trapezoid $ABCD,$ leg $\overline{BC}$ is perpendicular to bases $\overline{AB}$ and $\overline{CD},$ and diagonals $\overline{AC}$ and $\overline{BD}$ are perpendicular.  Given that $AB=\sqrt{11}$ and $AD=\sqrt{1001},$ find $BC^2.$
1. **Identify the given information and setup the problem:**
   - Trapezoid \(ABCD\) with \( \overline{BC} \) perpendicular to bases \( \overline{AB} \) and \( \overline{CD} \).
   - Diagonals \( \overline{AC} \) and \( \overline{BD} \) are perpendicular.
   - Given \( AB = \sqrt{11} \) and \( AD = \sqrt{1001} \).

2. **Introduce the points and angles:**
   - Let the perpendicular from \( A \) to \( CD \) intersect \( BD \) at \( F \).
   - Let the diagonals intersect at \( E \).
   - Let \( \angle EAF = \theta \).

3. **Establish the similarity of triangles:**
   - Since \( \angle EAF \cong \angle EBA \cong \angle ECB \cong \angle EDC = \theta \), the triangles \( \triangle EAF \), \( \triangle EBA \), \( \triangle ECB \), and \( \triangle EDC \) are similar by AA similarity (angle-angle similarity).

4. **Define the segments:**
   - Let \( AE = a \), \( BE = b \), \( CE = c \), and \( DE = d \).

5. **Set up the ratios from the similar triangles:**
   - From the similarity, we have:
     \[
     \frac{EF}{a} = \frac{a}{b} = \frac{b}{c} = \frac{c}{d}
     \]
   - Let the common ratio be \( x \), so \( \frac{b}{a} = x \).

6. **Express the given lengths in terms of \( a \) and \( x \):**
   - Using the Pythagorean theorem in the right triangles formed by the diagonals:
     \[
     a^2 + d^2 = a^2 + a^2 x^6 = a^2 (1 + x^6) = a^2 (1 + x^2)(1 - x^2 + x^4) = 1001
     \]
     \[
     a^2 + b^2 = a^2 + a^2 x^2 = a^2 (1 + x^2) = 11
     \]

7. **Divide the equations to find \( x \):**
   - Dividing the first equation by the second:
     \[
     \frac{a^2 (1 + x^6)}{a^2 (1 + x^2)} = \frac{1001}{11}
     \]
     \[
     1 + x^6 = \frac{1001}{11} (1 + x^2)
     \]
     \[
     1 + x^6 = 91 (1 + x^2)
     \]
     \[
     1 + x^6 = 91 + 91 x^2
     \]
     \[
     x^6 - 91 x^2 + 90 = 0
     \]

8. **Solve the quadratic equation in terms of \( y = x^2 \):**
   - Let \( y = x^2 \), then:
     \[
     y^3 - 91 y + 90 = 0
     \]
   - Using the quadratic formula for \( y \):
     \[
     y = \frac{91 \pm \sqrt{91^2 - 4 \cdot 90}}{2}
     \]
     \[
     y = \frac{91 \pm \sqrt{8281 - 360}}{2}
     \]
     \[
     y = \frac{91 \pm \sqrt{7921}}{2}
     \]
     \[
     y = \frac{91 \pm 89}{2}
     \]
     \[
     y = 90 \quad \text{or} \quad y = 1
     \]

9. **Determine the correct value of \( x \):**
   - Since \( y = x^2 \), we have \( x^2 = 10 \) (since \( x^2 = 1 \) does not satisfy the given conditions).

10. **Calculate \( BC^2 \):**
    - Since \( BC^2 = AB^2 x^2 \):
      \[
      BC^2 = (\sqrt{11})^2 \cdot 10 = 11 \cdot 10 = 110
      \]

The final answer is \( \boxed{110} \)