How to stop spawning enemies after countdown timer reaches zero?

I think you just have to call public void StopSpawning() then stop the timer... Try this :

  using System.Collections;
  using System.Collections.Generic;
  using UnityEngine;
  using UnityEngine.UI;
  
  public class Countdown_Timer : MonoBehaviour
  {
      float currentTime = 0f;
      float startingTime = 10f;
      [SerializeField] Text countdownText;
      private bool HasReached;
      public EnemySpawner Spawner; //Assign this in the inspector
      
      void Start()
      {
          HasReached = false;
          currentTime = startingTime;
      }
  
      void Update()
      {
          if (HasReached == false)
          {
              currentTime -= 1 * Time.deltaTime; //timer decreases by one each second
              countdownText.text = currentTime.ToString("0"); //conver float to string
  
              //stop the timer to zero                     
              if (currentTime <= 0) 
              {
                  HasReached = true;
                  Spawner.StopSpawning();                 
              }
          }         
      }
  }

Then for the EnemySpawer script :

  using System.Collections;
  using System.Collections.Generic;
  using UnityEngine;
  
  public class EnemySpawner : MonoBehaviour
  {
      public GameObject enemy;
      float randX;
      Vector2 whereToSpawn;
      public float spawnRate = 5f;
      float nextSpawn = 0.0f;
      bool spawn = true;
      bool HasSpawnedBoss;
  
      void Start()
      {
          HasSpawnedBoss = false;
      }
  
      void Update()
      {
          if (Time.time > nextSpawn && spawn == true)
          {
              nextSpawn = Time.time + spawnRate;
              randX = Random.Range(-18f, 18f);
              whereToSpawn = new Vector2(randX, transform.position.y);
              Instantiate(enemy, whereToSpawn, Quaternion.identity);
          }
          else if (spawn == false && HasSpawnedBoss == false)
          {
              HasSpawnedBoss = true;
              //Add your boss spawning script here
          }
      }
 
      public void StopSpawning()
      {
          spawn = false;
      }
  }