## Step 1: Define the variables and their attributes
We have four variables: 'apple pies', 'cornichons', 'protein bars', and 'slices of pizza'. Let's denote them as $x_0$, $x_1$, $x_2$, and $x_3$ respectively. The attributes or resources for these variables are given as:
- $r_0$: grams of protein, with $x_0$ having 10 grams, $x_1$ having 10 grams, $x_2$ having 9 grams, and $x_3$ having 9 grams.
- $r_1$: dollar cost, with $x_0$ costing 8 dollars, $x_1$ costing 11 dollars, $x_2$ costing 7 dollars, and $x_3$ costing 6 dollars.
- $r_2$: healthiness rating, with $x_0$ having a rating of 5, $x_1$ having a rating of 4, $x_2$ having a rating of 4, and $x_3$ having a rating of 1.

## 2: Formulate the objective function
The objective function to maximize is:
\[9x_0^2 + 3x_0x_1 + x_0x_3 + 8x_1x_2 + x_2^2 + 9x_3^2 + 7x_1 + 3x_2 + 2x_3\]

## 3: Define the constraints
### Protein Constraints
- $10x_0 + 10x_1 + 9x_2 + 9x_3 \geq 14$
- $10x_1 + 9x_2 + 9x_3 \geq 14$
- $10x_0 + 9x_2 + 9x_3 \geq 14$
- $10x_0 + 10x_1 + 9x_3 \geq 14$
- $10x_1 + 9x_2 + 9x_3 \geq 21$
- $10x_0 + 9x_2 + 9x_3 \geq 21$
- $10x_0 + 10x_1 + 9x_3 \geq 21$
- $10x_1 + 9x_2 + 9x_3 \geq 21$
- $10x_0 + 9x_2 + 9x_3 \geq 21$
- $10x_0 + 10x_1 + 9x_3 \geq 21$

### Cost Constraints
- $7x_2^2 + 6x_3^2 \geq 19$
- $8x_0 + 7x_2 \geq 14$
- $11x_1 + 7x_2 + 6x_3 \geq 21$
- $8x_0 + 11x_1 \leq 77$
- $11x_1 + 6x_3 \leq 28$
- $8^2x_0^2 + 11^2x_1^2 \leq 58$
- $7^2x_2^2 + 6^2x_3^2 \leq 81$
- $8x_0 + 11x_1 + 7x_2 + 6x_3 \leq 81$

### Healthiness Rating Constraints
- $4x_2 + x_3 \geq 13$
- $4x_1 + 4x_2 \leq 78$
- $4x_2 + x_3 \leq 43$
- $5x_0 + x_3 \leq 63$
- $4x_1 + x_3 \leq 25$
- $5^2x_0^2 + 4^2x_2^2 \leq 85$
- $5^2x_0^2 + 4^2x_2^2 + x_3^2 \leq 68$
- $4x_1 + 4x_2 + x_3 \leq 65$
- $5^2x_0^2 + 4^2x_1^2 + x_3^2 \leq 71$
- $5x_0 + 4x_1 + 4x_2 + x_3 \leq 71$

### Protein Squares Constraints
- $9^2x_2^2 + 9^2x_3^2 \leq 56$
- $10^2x_0^2 + 10^2x_1^2 \leq 63$
- $10^2x_1^2 + 9^2x_2^2 \leq 53$
- $10x_0 + 10x_1 + 9x_2 + 9x_3 \leq 53$

## 4: Implement the problem in Gurobi
```python
import gurobi as gp

# Define the model
m = gp.Model()

# Define the variables
x0 = m.addVar(name="apple_pies", lb=0)  # Non-integer amount allowed
x1 = m.addVar(name="cornichons", lb=0)  # Non-integer amount allowed
x2 = m.addVar(name="protein_bars", lb=0, integrality=gp.GRB.INTEGER)  # Integer amount required
x3 = m.addVar(name="slices_of_pizza", lb=0)  # Non-integer amount allowed

# Objective function
m.setObjective(9*x0**2 + 3*x0*x1 + x0*x3 + 8*x1*x2 + x2**2 + 9*x3**2 + 7*x1 + 3*x2 + 2*x3, gp.GRB.MAXIMIZE)

# Constraints
# Protein constraints
m.addConstr(10*x0 + 10*x1 + 9*x2 + 9*x3 >= 14)
m.addConstr(10*x1 + 9*x2 + 9*x3 >= 14)
m.addConstr(10*x0 + 9*x2 + 9*x3 >= 14)
m.addConstr(10*x0 + 10*x1 + 9*x3 >= 14)
m.addConstr(10*x1 + 9*x2 + 9*x3 >= 21)
m.addConstr(10*x0 + 9*x2 + 9*x3 >= 21)
m.addConstr(10*x0 + 10*x1 + 9*x3 >= 21)
m.addConstr(10*x1 + 9*x2 + 9*x3 >= 21)
m.addConstr(10*x0 + 9*x2 + 9*x3 >= 21)
m.addConstr(10*x0 + 10*x1 + 9*x3 >= 21)

# Cost constraints
m.addConstr(7*x2**2 + 6*x3**2 >= 19)
m.addConstr(8*x0 + 7*x2 >= 14)
m.addConstr(11*x1 + 7*x2 + 6*x3 >= 21)
m.addConstr(8*x0 + 11*x1 <= 77)
m.addConstr(11*x1 + 6*x3 <= 28)
m.addConstr(8**2*x0**2 + 11**2*x1**2 <= 58)
m.addConstr(7**2*x2**2 + 6**2*x3**2 <= 81)
m.addConstr(8*x0 + 11*x1 + 7*x2 + 6*x3 <= 81)

# Healthiness rating constraints
m.addConstr(4*x2 + x3 >= 13)
m.addConstr(4*x1 + 4*x2 <= 78)
m.addConstr(4*x2 + x3 <= 43)
m.addConstr(5*x0 + x3 <= 63)
m.addConstr(4*x1 + x3 <= 25)
m.addConstr(5**2*x0**2 + 4**2*x2**2 <= 85)
m.addConstr(5**2*x0**2 + 4**2*x2**2 + x3**2 <= 68)
m.addConstr(4*x1 + 4*x2 + x3 <= 65)
m.addConstr(5**2*x0**2 + 4**2*x1**2 + x3**2 <= 71)
m.addConstr(5*x0 + 4*x1 + 4*x2 + x3 <= 71)

# Protein squares constraints
m.addConstr(9**2*x2**2 + 9**2*x3**2 <= 56)
m.addConstr(10**2*x0**2 + 10**2*x1**2 <= 63)
m.addConstr(10**2*x1**2 + 9**2*x2**2 <= 53)
m.addConstr(10*x0 + 10*x1 + 9*x2 + 9*x3 <= 53)

# Solve the model
m.optimize()

# Print the solution
if m.status == gp.GRB.OPTIMAL:
    print("Objective: ", m.objVal)
    print("Apple pies: ", x0.varValue)
    print("Cornichons: ", x1.varValue)
    print("Protein bars: ", x2.varValue)
    print("Slices of pizza: ", x3.varValue)
else:
    print("The model is infeasible")
```