To solve the given optimization problem, we first need to translate the natural language description into a mathematical formulation that can be implemented in Gurobi.

### Mathematical Formulation

Let's denote:
- \(x_0\) as the number of squash plants,
- \(x_1\) as the number of chives,
- \(x_2\) as the number of orange trees.

The objective function to minimize is:
\[ 7x_0^2 + 9x_0x_1 + 2x_2^2 \]

Subject to the constraints:
1. \( 23x_0 + 16x_1 \geq 129 \)
2. \( 23^2x_0^2 + 13^2x_2^2 \geq 84 \)
3. \( 23x_0 + 16x_1 + 13x_2 \geq 84 \)
4. \( 9x_1 - 3x_2 \geq 0 \)
5. \( 5x_0 - 7x_2 \geq 0 \)
6. \( 23x_0 + 16x_1 \leq 314 \)
7. \( x_0 \in \mathbb{Z} \) (integer, but Gurobi will handle this with `VarType` or `integrality` constraint)
8. \( x_1 \in \mathbb{Z} \) (integer)
9. \( x_2 \in \mathbb{Z} \) (integer)

### Gurobi Code

```python
import gurobi

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

    # Define variables
    x0 = model.addVar(name="squash_plants", vtype=gurobi.GRB.INTEGER)
    x1 = model.addVar(name="chives", vtype=gurobi.GRB.INTEGER)
    x2 = model.addVar(name="orange_trees", vtype=gurobi.GRB.INTEGER)

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

    # Constraints
    model.addConstr(23*x0 + 16*x1 >= 129, name="beauty_rating_squash_chives")
    model.addConstr(23**2*x0**2 + 13**2*x2**2 >= 84, name="beauty_rating_squash_orange")
    model.addConstr(23*x0 + 16*x1 + 13*x2 >= 84, name="total_beauty_rating")
    model.addConstr(9*x1 - 3*x2 >= 0, name="chives_orange_trees")
    model.addConstr(5*x0 - 7*x2 >= 0, name="squash_orange_trees")
    model.addConstr(23*x0 + 16*x1 <= 314, name="max_beauty_rating_squash_chives")

    # Solve the model
    model.optimize()

    # Print the solution
    if model.status == gurobi.GRB.OPTIMAL:
        print("Objective: ", model.objval)
        print("Squash Plants: ", x0.varValue)
        print("Chives: ", x1.varValue)
        print("Orange Trees: ", x2.varValue)
    else:
        print("The model is infeasible")

solve_optimization_problem()
```

This code defines the optimization problem as per the given description and solves it using Gurobi. Note that Gurobi handles variable types (integer, continuous, etc.) through the `vtype` parameter when adding variables to the model. The `GRB.INTEGER` type is used for all variables to ensure they are treated as integers. 

The model is then solved, and if an optimal solution is found, it prints out the objective function value and the values of the variables. If the model is infeasible, it indicates so.