I don't have it black but grey (with a blue rect).
Also i'd suggest to keep variables name like width and height reserved for processing for code readability.
You could append something for your class fields and constructor arguments like

1. class ExcludeArea {
2. 
   
3.   int areaWidth, areaHeight;
4.   PShape excludeShape;
5.   PGraphics excludeCheckArea;
6. 
   
7.   ExcludeArea(int lwidth, int lheight, String excludeShape) {
8.     areaWidth = lwidth;
9.     areaHeight = lheight;
10.     this.excludeShape = loadShape(excludeShape);
11.     excludeCheckArea = createGraphics(width, height, P2D);
12.     createExcludeArea();
13.   }
14. ...
    

how can i get a white background with a red rectangle for example?

Replace 100 with 255 in the background() call.

i know phi.lho but try this:
black background...

1. ExcludeArea ex;
   
   void setup() {
     size(1000, 1000);
     ex = new ExcludeArea(width, height);
     exit();
   }
   
   
   
   
   class ExcludeArea {
   
     int width, height;
   
     PShape excludeShape;
     PGraphics excludeCheckArea;
   
     ExcludeArea(int width, int height) {
       this.width = width;
       this.height = height;
       excludeCheckArea = createGraphics(width, height, P2D);
   
       createExcludeArea();
     }
   
     void createExcludeArea() {
   
       colorMode(RGB, 255, 255, 255);
       excludeCheckArea.beginDraw();
       excludeCheckArea.colorMode(RGB, 255, 255, 255);
       excludeCheckArea.background(255);
       excludeCheckArea.fill(0, 0, 0);
       excludeCheckArea.rect(0, 0, width/2, height/2);
       excludeCheckArea.endDraw();
       excludeCheckArea.save("excludeCheckArea.jpg");
     }
   }
   

I just tried your code and for me the background isn't any black but more red.

deja vu.

I don't need it anymore but whenever i need i will try the next processing version.

thanks