## Step 1: Define the symbolic representation of the variables
The variables are 'cheeseburgers', 'tomatoes', and 'milkshakes', which can be represented symbolically as $x_0$, $x_1$, and $x_2$ respectively.

## Step 2: Convert the objective function into a symbolic representation
The objective function to maximize is $1 \cdot x_0^2 + 1 \cdot x_0 \cdot x_2 + 9 \cdot x_1^2 + 4 \cdot x_2$.

## Step 3: List the constraints in symbolic notation
The constraints are:
- $12x_0 + x_1 + 11x_2 \leq 66$ (milligrams of calcium)
- $7x_0 + 6x_1 + 2x_2 \leq 55$ (grams of fiber)
- $2x_0 + 3x_1 + 9x_2 \leq 62$ (grams of fat)
- $6x_1 + 2x_2 \geq 15$ (at least 15 grams of fiber from tomatoes and milkshakes)
- $x_1 + 11x_2 \leq 46$ (up to 46 milligrams of calcium from tomatoes and milkshakes)
- $12x_0 + x_1 \leq 25$ (at most 25 milligrams of calcium from cheeseburgers and tomatoes)
- $12x_0 + x_1 + 11x_2 \leq 25$ (at most 25 milligrams of calcium from all sources)
- $7x_0 + 6x_1 \leq 23$ (no more than 23 grams of fiber from cheeseburgers and tomatoes)
- $7x_0 + 2x_2 \leq 41$ (no more than 41 grams of fiber from cheeseburgers and milkshakes)
- $7^2x_0^2 + 6^2x_1^2 + 2^2x_2^2 \leq 21^2$ (at most 21 grams of fiber from squared sources)
- $7x_0 + 6x_1 + 2x_2 \leq 21$ (at most 21 grams of fiber from all sources)
- $2x_0 + 3x_1 \leq 44$ (no more than 44 grams of fat from cheeseburgers and tomatoes)
- $3^2x_1^2 + 9^2x_2^2 \leq 20^2$ (up to 20 grams of fat from squared tomatoes and milkshakes)
- $2^2x_0^2 + 3^2x_1^2 + 9^2x_2^2 \leq 28^2$ (no more than 28 grams of fat from squared sources)
- $2x_0 + 3x_1 + 9x_2 \leq 28$ (at most 28 grams of fat from all sources)
- $x_0 \geq 0$ and $x_0$ is an integer (non-fractional amount of cheeseburgers)
- $x_1 \geq 0$ (non-negative amount of tomatoes)
- $x_2 \geq 0$ and $x_2$ is an integer (integer number of milkshakes)

## Step 4: Provide the symbolic representation in JSON format
```json
{
    'sym_variables': [('x0', 'cheeseburgers'), ('x1', 'tomatoes'), ('x2', 'milkshakes')],
    'objective_function': '1*x0^2 + 1*x0*x2 + 9*x1^2 + 4*x2',
    'constraints': [
        '12*x0 + x1 + 11*x2 <= 66',
        '7*x0 + 6*x1 + 2*x2 <= 55',
        '2*x0 + 3*x1 + 9*x2 <= 62',
        '6*x1 + 2*x2 >= 15',
        'x1 + 11*x2 <= 46',
        '12*x0 + x1 <= 25',
        '12*x0 + x1 + 11*x2 <= 25',
        '7*x0 + 6*x1 <= 23',
        '7*x0 + 2*x2 <= 41',
        '49*x0^2 + 36*x1^2 + 4*x2^2 <= 441',
        '7*x0 + 6*x1 + 2*x2 <= 21',
        '2*x0 + 3*x1 <= 44',
        '9*x1^2 + 81*x2^2 <= 400',
        '4*x0^2 + 9*x1^2 + 81*x2^2 <= 784',
        '2*x0 + 3*x1 + 9*x2 <= 28',
        'x0 >= 0 and x0 is an integer',
        'x1 >= 0',
        'x2 >= 0 and x2 is an integer'
    ]
}
```

## Step 5: Write the Gurobi code to solve the optimization problem
```python
import gurobi

# Create a new model
m = gurobi.Model()

# Define the variables
x0 = m.addVar(name="cheeseburgers", vtype=gurobi.GRB.INTEGER, lb=0)
x1 = m.addVar(name="tomatoes", lb=0)
x2 = m.addVar(name="milkshakes", vtype=gurobi.GRB.INTEGER, lb=0)

# Objective function
m.setObjective(x0**2 + x0*x2 + 9*x1**2 + 4*x2, gurobi.GRB.MAXIMIZE)

# Constraints
m.addConstr(12*x0 + x1 + 11*x2 <= 66, name="calcium")
m.addConstr(7*x0 + 6*x1 + 2*x2 <= 55, name="fiber")
m.addConstr(2*x0 + 3*x1 + 9*x2 <= 62, name="fat")
m.addConstr(6*x1 + 2*x2 >= 15, name="fiber_from_tomatoes_and_milkshakes")
m.addConstr(x1 + 11*x2 <= 46, name="calcium_from_tomatoes_and_milkshakes")
m.addConstr(12*x0 + x1 <= 25, name="calcium_from_cheeseburgers_and_tomatoes")
m.addConstr(12*x0 + x1 + 11*x2 <= 25, name="total_calcium")
m.addConstr(7*x0 + 6*x1 <= 23, name="fiber_from_cheeseburgers_and_tomatoes")
m.addConstr(7*x0 + 2*x2 <= 41, name="fiber_from_cheeseburgers_and_milkshakes")
m.addConstr(49*x0**2 + 36*x1**2 + 4*x2**2 <= 441, name="fiber_squared")
m.addConstr(7*x0 + 6*x1 + 2*x2 <= 21, name="total_fiber")
m.addConstr(2*x0 + 3*x1 <= 44, name="fat_from_cheeseburgers_and_tomatoes")
m.addConstr(9*x1**2 + 81*x2**2 <= 400, name="fat_squared_tomatoes_milkshakes")
m.addConstr(4*x0**2 + 9*x1**2 + 81*x2**2 <= 784, name="fat_squared")
m.addConstr(2*x0 + 3*x1 + 9*x2 <= 28, name="total_fat")

# Optimize the model
m.optimize()

# Print the solution
if m.status == gurobi.GRB.OPTIMAL:
    print("Objective: ", m.objVal)
    print("Cheeseburgers: ", x0.varValue)
    print("Tomatoes: ", x1.varValue)
    print("Milkshakes: ", x2.varValue)
else:
    print("The model is infeasible")
```