Cannot click on a button while Drag Events are called

Good question - it does seem like this should be an included functionality in the UI system - but it just doesn't seem to support both events used together. To remedy this, we'll create our own button component that will completely replace the existing button. (Note: you can still use the existing button component alongside this if you'd like).

We'll need to use Unity's Events and EventSystem libraries. To do this, just put these two new lines at the top of your script!

     using UnityEngine.Events;
     using UnityEngine.EventSystems;

Next we'll need to add two Interfaces. These will allow us to listen to Unity's event system (the same one that you're using when you add a Button or EventTrigger component). We want to listen to two of them in particular - called IPointerDownHandler and IPointerUpHandler.

You can add them by typing a comma after Monobehavior (or whatever other class you're inheriting from) and then typing every interface you need. Just start typing IPointer... and your auto-complete options will show you the rest. Check them out!

  public class CustomButton : MonoBehaviour, IPointerDownHandler, IPointerUpHandler

Now that we've added our interfaces, we can implement the methods that they require. In a nutshell, interfaces add specific functionality (like listening for events from an event system). However, you have to agree to implement the methods that they want you to - or it won't compile.

When you implement these methods, they have to share the same Access Modifiers, method name, and parameters.

The IPointerDownHandler interface wants us to use a public method called OnPointerDown. In that method, it wants us to have a PointerEventData parameter. So lets do that.

 public void OnPointerDown(PointerEventData data)
     {
         OnButtonDown.Invoke();
     }

And a similar case with IPointerUpHandler

     public void OnPointerUp(PointerEventData data)
     {
         OnButtonUp.Invoke();
     }

As you can see I'm just invoking our stored methods when these are called from Unity's event system.

And that's all there is to it! Full script below for yours and others' convenience.

     using UnityEngine;
     using UnityEngine.Events;
     using UnityEngine.EventSystems;
     
     public class CustomButton : MonoBehaviour, IPointerDownHandler, IPointerUpHandler
     {
         //Called when we click the button down
         public UnityEvent OnButtonDown;
     
         //Called when we click the button up
         public UnityEvent OnButtonUp;
     
         public void OnPointerDown(PointerEventData data)
         {
             OnButtonDown.Invoke();
         }
     
         public void OnPointerUp(PointerEventData data)
         {
             OnButtonUp.Invoke();
         }
     }