Need help with my first script, changing cube colors

OnMouseOver() means the mouse is hovering over the object. You're checking to see if the mouse button is pressed, which it might not be.

Ah that makes sense, well now I did this and the first two work now, the cube goes blue and black, but apparently the compiler is telling me that I cant reuse OnMouseDown and OnMouseUp... any ideas? I'm thinking maybe MonoBehaviour may be a reason cosidering I'm trying to get it to do more than one thing?

 using UnityEngine;
 using System.Collections;
 
 public class Block_Start : MonoBehaviour {
 
     void OnMouseDown () 
     {
         Debug.Log("Mouse is Down");
         renderer.material.color = Color.black;
 }
         
     void OnMouseUp() 
     {
             Debug.Log("Mouse is Up");
             renderer.material.color = Color.red;
         }
 
     void OnMouseDown () 
     {
         
                 if (Input.GetMouseButtonDown (1))
                         Debug.Log ("Secondary Mouse is Down");
                 renderer.material.color = Color.green;
 }
 
     void OnMouseUp()
     {
                 if (Input.GetMouseButtonUp (1))
                         Debug.Log ("Secondary Mouse is Up");
                 renderer.material.color = Color.blue;
 }
 
     void OnMouseDown()
     {
                 if (Input.GetMouseButtonDown (2))
                         Debug.Log ("Back to Default");
                 renderer.material.color = Color.white;
     }
 
 }

Your original thinking was sound enough, see my answer. You can't reuse the same name for a method unless you overload it. The OnMouseUp/Down/etc methods are basically the same as any other method (function) you might write. Except in this case unity will invoke it if a particular mouse event occurs.

MonoBehaviour is simply the base class that all unity scripts derive from. In C# a colon is used to show that a particular class derives (inherits) from another. You can take away MonoBehaviour but none of the unity methods or properties will be available.