To solve the given optimization problem using Gurobi, we need to translate the natural language description into a mathematical model. The objective function and constraints can be defined as follows:

- **Objective Function**: Minimize \(5x_0^2 + 9x_0x_1 + 3x_1^2 + 6x_0 + 4x_1\), where \(x_0\) represents the quantity of peanut butter sandwiches and \(x_1\) represents the quantity of milkshakes.

- **Constraints**:
  - Fiber from peanut butter sandwiches and milkshakes: \(0.83x_0 + 0.68x_1 \leq 143\)
  - Tastiness rating constraint: \(6.28x_0 + 10.37x_1 \geq 18\) and \(6.28x_0 + 10.37x_1 \leq 73\)
  - Minimum fiber requirement from both sources: \(0.83x_0 + 0.68x_1 \geq 55\)
  - Minimum fiber squared requirement (interpreted as the sum of squares for simplicity, but note this might not be the intended interpretation without further clarification): \(0.83^2x_0^2 + 0.68^2x_1^2 \geq 55^2\)
  - Linear constraint: \(3x_0 - x_1 \geq 0\)
  - Maximum fiber constraint: \(0.83x_0 + 0.68x_1 \leq 100\)

Given the complexity and potential inconsistencies in interpreting some constraints (like the "fiber squared" requirement), we'll proceed with a direct translation assuming all constraints are meant to be applied as written, focusing on providing a Gurobi model that captures these requirements.

```python
from gurobipy import *

# Create a new model
m = Model("Optimization_Problem")

# Define variables
x0 = m.addVar(lb=0, ub=GRB.INFINITY, vtype=GRB.CONTINUOUS, name="peanutbutter_sandwiches")
x1 = m.addVar(lb=0, ub=GRB.INFINITY, vtype=GRB.CONTINUOUS, name="milkshakes")

# Objective function
m.setObjective(5*x0**2 + 9*x0*x1 + 3*x1**2 + 6*x0 + 4*x1, GRB.MINIMIZE)

# Constraints
m.addConstr(0.83*x0 + 0.68*x1 <= 143, name="fiber_constraint")
m.addConstr(6.28*x0 + 10.37*x1 >= 18, name="tastiness_min")
m.addConstr(6.28*x0 + 10.37*x1 <= 73, name="tastiness_max")
m.addConstr(0.83*x0 + 0.68*x1 >= 55, name="min_fiber")
m.addConstr((0.83**2)*x0**2 + (0.68**2)*x1**2 >= 55**2, name="min_fiber_squared")
m.addConstr(3*x0 - x1 >= 0, name="linear_constraint")
m.addConstr(0.83*x0 + 0.68*x1 <= 100, name="max_fiber")

# Optimize model
m.optimize()

# Print solution
if m.status == GRB.OPTIMAL:
    print("Optimal solution found:")
    print(f"peanutbutter_sandwiches: {x0.x}")
    print(f"milkshakes: {x1.x}")
    print(f"Objective function value: {m.objVal}")
else:
    print("No optimal solution found")
```