Some small fixes: in general, when you draw something that must not be persistent (most of the time), you must put a background() call at the start of draw(). Otherwise, any animation will leave a trail and fixed stuff will get blocky.
Like that:

1. void draw() {
2.   background(255);
3.  
4.   //draw blinds 
5.   stroke(1);
6.   strokeWeight(1);
7.   y = 0;
8.   while (y < height) {
9.     line(0, y, width, y); 
10.     y = y + 5;
11.   }
12.  
13.   //draw string
14.   line(150, 60, 150, 0);
15.  
16.   //draw button at end of string
17.   rectMode(CENTER);
18.   fill(0);
19.   rect(150, 60, 10, 10);
20. }

Notice I have reset y, and I put the state changes (stroke, fill...) before the drawing orders.
For dragging a shape, there are several example sketches illustrating this (see the Learning section of the site).
You need a couple of variables to track the rectangle's position, and once the mouse click within the rectangle has been detected, you should have a boolean variable indicating you are dragging the rect, otherwise any fast move will put the mouse cursor outside of the rect and the dragging will cease.
Ie. start dragging when in the rect, don't use this condition when dragging. And detect the mouse release to go out of this state.

more like (using mouse built in functions)