## Step 1: Define the optimization problem
The problem is to maximize the objective function: $1 \times \text{mg of vitamin B3} \times \text{mg of calcium} + 9 \times (\text{mg of calcium})^2 + 4 \times \text{mg of vitamin B3} + 3 \times \text{mg of calcium}$.

## Step 2: Identify the variables
Let $x_0$ be the milligrams of vitamin B3 and $x_1$ be the milligrams of calcium.

## 3: Formulate the objective function
The objective function to maximize is: $x_0x_1 + 9x_1^2 + 4x_0 + 3x_1$.

## 4: Define the constraints
1. The muscle growth index of $x_0$ is 3: $3x_0 = 3$.
2. The cardiovascular support index of $x_0$ is 1: $1x_0 = 1$.
3. The digestive support index for $x_0$ is 8: $8x_0 = 8$.
4. The muscle growth index for $x_1$ is 3: $3x_1 = 3$.
5. The cardiovascular support index for $x_1$ is 4: $4x_1 = 4$.
6. The digestive support index for $x_1$ is 11: $11x_1 = 11$.
7. Total combined muscle growth index: $3x_0 + 3x_1 \geq 20$.
8. Total combined cardiovascular support index: $1x_0^2 + 4x_1^2 \geq 56$.
9. Total combined digestive support index: $8x_0 + 11x_1 \geq 61$.
10. $6x_0^2 - 8x_1^2 \geq 0$.
11. Total combined muscle growth index: $3x_0 + 3x_1 \leq 66$.
12. Total combined cardiovascular support index: $1x_0^2 + 4x_1^2 \leq 74$.
13. Total combined digestive support index: $8x_0 + 11x_1 \leq 95$.
14. $x_0$ is an integer.
15. $x_1$ can be a fraction.

## 5: Correct and simplify constraints
From constraints 1, 2, and 3, we get $x_0 = 1$ (as they all imply $x_0 = 1$).
From constraints 4, 5, and 6, we get $x_1 = 1$ (as they all imply $x_1 = 1$).

However, we need to consider all constraints for completeness and accuracy in formulating the problem for Gurobi.

## 6: Write the Gurobi code
Given the specific nature of the problem and the need for precise formulation, let's correct and directly formulate the problem with Gurobi.

```python
import gurobi

def optimize_problem():
    # Create a new model
    model = gurobi.Model()

    # Define variables
    x0 = model.addVar(name="milligrams of vitamin B3", vtype=gurobi.GRB.INTEGER)
    x1 = model.addVar(name="milligrams of calcium")

    # Objective function
    model.setObjective(x0 * x1 + 9 * x1**2 + 4 * x0 + 3 * x1, gurobi.GRB.MAXIMIZE)

    # Constraints
    model.addConstr(x0 == 1, name="muscle_growth_index_vitamin_B3")
    model.addConstr(x1 == 1, name="cardiovascular_support_index_calcium") # This was incorrectly stated, using 1*x0 = 1 and 4*x1 = 4 implies x0 = 1 and x1 = 1 directly

    # However, let's properly account for all given constraints in a more accurate mathematical representation
    model.addConstr(3*x0 + 3*x1 >= 20, name="total_muscle_growth_index")
    model.addConstr(x0**2 + 4*x1**2 >= 56, name="total_cardiovascular_support_index")
    model.addConstr(8*x0 + 11*x1 >= 61, name="total_digestive_support_index")
    model.addConstr(6*x0**2 - 8*x1**2 >= 0, name="quadratic_constraint")
    model.addConstr(3*x0 + 3*x1 <= 66, name="total_muscle_growth_index_upper_bound")
    model.addConstr(x0**2 + 4*x1**2 <= 74, name="total_cardiovascular_support_index_upper_bound")
    model.addConstr(8*x0 + 11*x1 <= 95, name="total_digestive_support_index_upper_bound")

    # Optimize
    model.optimize()

    # Print solution
    if model.status == gurobi.GRB.OPTIMAL:
        print("Optimal solution found.")
        print(f"Milligrams of vitamin B3: {x0.varValue}")
        print(f"Milligrams of calcium: {x1.varValue}")
        print(f"Objective: {model.objVal}")
    else:
        print("No optimal solution found.")

optimize_problem()
```

The final answer is: 
```python
import gurobi

def optimize_problem():
    model = gurobi.Model()

    x0 = model.addVar(name="milligrams of vitamin B3", vtype=gurobi.GRB.INTEGER)
    x1 = model.addVar(name="milligrams of calcium")

    model.setObjective(x0 * x1 + 9 * x1**2 + 4 * x0 + 3 * x1, gurobi.GRB.MAXIMIZE)

    model.addConstr(x0 == 1, name="constraint_x0")
    model.addConstr(3*x0 + 3*x1 >= 20, name="total_muscle_growth_index")
    model.addConstr(x0**2 + 4*x1**2 >= 56, name="total_cardiovascular_support_index")
    model.addConstr(8*x0 + 11*x1 >= 61, name="total_digestive_support_index")
    model.addConstr(6*x0**2 - 8*x1**2 >= 0, name="quadratic_constraint")
    model.addConstr(3*x0 + 3*x1 <= 66, name="total_muscle_growth_index_upper_bound")
    model.addConstr(x0**2 + 4*x1**2 <= 74, name="total_cardiovascular_support_index_upper_bound")
    model.addConstr(8*x0 + 11*x1 <= 95, name="total_digestive_support_index_upper_bound")

    model.optimize()

    if model.status == gurobi.GRB.OPTIMAL:
        print("Optimal solution found.")
        print(f"Milligrams of vitamin B3: {x0.varValue}")
        print(f"Milligrams of calcium: {x1.varValue}")
        print(f"Objective: {model.objVal}")
    else:
        print("No optimal solution found.")

optimize_problem()
```