int x = 0;  // x location of square
int y = 0;  // y location of square

int speed = 5;  // speed of square

int state = 0;  // "state" of square, it's state will determine how it moves

void setup() {
 size(200,200);
}

void draw() {
 framerate(30);
 background(0);
 noStroke();
 fill(255);

 // display the square
 rect(x,y,10,10);

 // Depending on the state, change location of the square
 // Depending on the location, change the state of square
 if (state == 0) {
   x+=speed;
   if (x > width-10) {
     x = width-10;
     state++;
   }
 }
 else if (state == 1) {
   y+= speed;
   if (y > height-10) {
     y = height-10;
     state++;
   }
 }
 else if (state == 2) {
   x-=speed;
   if (x < 0) {
     x = 0;
     state++;
   }
 }
 else if (state == 3) {
   y-=speed;
   if (y < 0) {
     y = 0;
     state=0;
   }
 }


}


Why does encapsulating the if statements into a
for(;x<width-10;x+=speed)
{
 rect(x,y,10,10);
delay(100);
background(0);
}
(other for loops omitted)

not work as expected?

The only reason I could see for putting that into a for loop is if you wanted to have multiple squares all moving independently...  In that case the problem is likely to be that you'd need multiple 'state' variables - one for each square.  If that's the problem then the solution is to create a square object with a state property and associated movement methods...

(But as the others pointed out - you haven't expressed your problem particularly well.)

This is code from Daniel Shiffman's website. The framerate was the only thing (I think) that was outdated and is easily fixed.

The code makes a square move around the border. What I want to do is animate the whole "move around the border" in one draw loop. But when I try, it breaks.

int x = 0;  // x location of square
int y = 0;  // y location of square

int speed = 5;  // speed of square

int state = 0;  // "state" of square, it's state will determine how it moves

void setup() {
 size(200,200);
}

void draw() {
 frameRate(30);
 background(0);
 noStroke();
 fill(255);

 // display the square
 //rect(x,y,10,10);

 // Depending on the state, change location of the square
 // Depending on the location, change the state of square
 for(;x<width-10;x+=speed)
   rect(x,y,10,10);
   delay(100);
   background(0);
 x = width-10;
 for(;y<height-10;y+=speed)
   rect(x,y,10,10);
   delay(100);
   background(0);
 y = height-10;
 for(;x>0;x-=speed)
   rect(x,y,10,10);
   delay(100);
   background(0);
 x = 0;
 for(;y>0;y-=speed)
   rect(x,y,10,10);
   delay(100);
   background(0);
 y = 0;
}