## Step 1: Define the optimization problem and identify the variables, objective function, and constraints.
The optimization problem involves minimizing a given objective function subject to several constraints. The variables are ['CBRN platoons', 'artillery batteries', 'water purification units', 'medical platoons'], denoted as $x_0, x_1, x_2, x_3$ respectively.

## 2: Formulate the objective function.
The objective function to minimize is $4x_0^2 + 7x_0x_1 + 9x_0x_2 + 7x_0x_3 + 3x_1^2 + 2x_1x_2 + 8x_1x_3 + 5x_2x_3 + 5x_3^2 + 3x_0 + x_2$.

## 3: Define the constraints based on the given information.
- $3x_0 = 3$ (logistics footprint of CBRN platoons)
- $6x_0 = 6$ (fun factor of CBRN platoons)
- $2x_1$ (logistics footprint of artillery batteries)
- $x_1$ (fun factor of artillery batteries)
- $5x_2$ (logistics footprint of water purification units)
- $6x_2$ (fun factor of water purification units)
- $7x_3$ (logistics footprint of medical platoons)
- $6x_3$ (fun factor of medical platoons)
- $5x_2 + 7x_3 \geq 17$ (total logistics footprint of water purification units and medical platoons)
- $3^2 + 5^2 \geq 23$ is a constant and always true, but we consider $3x_0 + 5x_2 \geq 23$ for variability (minimum logistics footprint of CBRN platoons squared and water purification units squared, adjusting for actual meaning)
- $3x_0 + 2x_1 \geq 11$ (total logistics footprint of CBRN platoons and artillery batteries)
- $2x_1 + 5x_2 \geq 19$ (total logistics footprint of artillery batteries and water purification units)
- $3x_0 + 2x_1 + 5x_2 + 7x_3 \geq 19$ (total logistics footprint of all units)
- $6x_0 + x_1 \geq 14$ (total combined fun factor from CBRN platoons and artillery batteries)
- $6x_2 + 6x_3 \geq 20$ (total combined fun factor from water purification units and medical platoons)
- $6x_0 + x_1 + 6x_2 + 6x_3 \geq 20$ (total combined fun factor from all units)
- $3x_0 + 2x_1 \leq 106$ (maximum logistics footprint from CBRN platoons and artillery batteries)
- $2^2x_1^2 + 7^2x_3^2 \leq 97$ (maximum logistics footprint from artillery batteries squared and medical platoons squared)
- $2^2x_1^2 + 5^2x_2^2 \leq 59$ (maximum logistics footprint from artillery batteries squared and water purification units squared)
- $3x_0 + 5x_2 + 7x_3 \leq 101$ (maximum logistics footprint from CBRN platoons, water purification units, and medical platoons)
- $x_0, x_1, x_2, x_3$ are integers.

## 4: Correct and simplify constraints for accurate modeling.
Given that $3x_0 = 3$ and $6x_0 = 6$, it implies $x_0 = 1$. 
- The fun factor and logistics footprint constraints for individual units are equality constraints but given $x_0=1$, they are satisfied.

## 5: Implement the problem in Gurobi Python.
```python
import gurobi

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

    # Define variables
    x0 = model.addVar(name="CBRN_platoons", vtype=gurobi.GRB.INTEGER)
    x1 = model.addVar(name="artillery_batteries", vtype=gurobi.GRB.INTEGER)
    x2 = model.addVar(name="water_purification_units", vtype=gurobi.GRB.INTEGER)
    x3 = model.addVar(name="medical_platoons", vtype=gurobi.GRB.INTEGER)

    # Objective function
    model.setObjective(4*x0**2 + 7*x0*x1 + 9*x0*x2 + 7*x0*x3 + 3*x1**2 + 2*x1*x2 + 8*x1*x3 + 5*x2*x3 + 5*x3**2 + 3*x0 + x2, gurobi.GRB.MINIMIZE)

    # Constraints
    model.addConstr(x0 == 1)  # Given
    model.addConstr(5*x2 + 7*x3 >= 17)
    model.addConstr(3*x0 + 5*x2 >= 23)  
    model.addConstr(3*x0 + 2*x1 >= 11)
    model.addConstr(2*x1 + 5*x2 >= 19)
    model.addConstr(3*x0 + 2*x1 + 5*x2 + 7*x3 >= 19)
    model.addConstr(6*x0 + x1 >= 14)
    model.addConstr(6*x2 + 6*x3 >= 20)
    model.addConstr(6*x0 + x1 + 6*x2 + 6*x3 >= 20)
    model.addConstr(3*x0 + 2*x1 <= 106)
    model.addConstr(4*x1**2 + 49*x3**2 <= 97)
    model.addConstr(4*x1**2 + 25*x2**2 <= 59)
    model.addConstr(3*x0 + 5*x2 + 7*x3 <= 101)

    # Optimize
    model.optimize()

    # Print solution
    if model.status == gurobi.GRB.OPTIMAL:
        print("Objective: ", model.objVal)
        print("CBRN platoons: ", x0.varValue)
        print("Artillery batteries: ", x1.varValue)
        print("Water purification units: ", x2.varValue)
        print("Medical platoons: ", x3.varValue)
    else:
        print("No solution found")

optimize_problem()
```