Certainly! Below is a Python function that generates test cases for the given problem statement:

```python
import random

def tcgen():
    # Define the constraints
    MIN_N = 2
    MAX_N = 10**6
    MOD = 10**9 + 7
    
    # Generate a random value for N within the given constraints
    N = random.randint(MIN_N, MAX_N)
    
    return N

# Example usage:
if __name__ == "__main__":
    # Generate a test case
    print(tcgen())
```

### Explanation
1. **Constraints**:
    - \( N \) ranges from 2 to \( 10^6 \).
   
2. **Function `tcgen`**:
    - Randomly selects an integer \( N \) within the given constraints.
    - Returns \( N \).

### Example Usage
Run the script and it will print a random integer \( N \) within the specified range, which represents the number of squares.

### Testing
To test the function, you can simply call `tcgen()` multiple times to see different generated values for \( N \). Make sure to integrate this with your solution to verify correctness with various input sizes.

```python
if __name__ == "__main__":
    for _ in range(5):  # Generate 5 test cases for demonstration
        print(tcgen())
```

This would give you a good spread of values for \( N \) to test against your solution for correctness and performance.