hm....

not sure if I understand the core of your question....

Is it this:

I'm a little confused about how to go about allowing the player to slide against the box.

Do you have this only when you go diagonally or always?



Anyway:

• OK, so when you go right, you only have to check the right side of the box (make an extra function collisionForTheRightSideOfThePlayer). When you detect a collision, set the player so that he touches the obstacle / box (e.g. player.x = box.x - player.width-1 or so). Set a var obstacleRight to true and check this var too.
• When you go left (or up / down) accordingly

see

http://wiki.processing.org/w/Rect-Rect_intersection

this:

1. int isColliding(Box box1, Box box2)
2. {
3.   if (!((box1.posX + box1.sizeX) >= box2.posX)) return 0;
4.   if (!(box1.posX <= (box2.posX + box2.sizeX))) return 0;
5.   if (!((box1.posY - box1.sizeY) <= box2.posY)) return 0;
6.   if (!(box1.posY >= (box2.posY - box2.sizeY))) return 0;
7.   
8.   return 1;
9. }

I didn't fully understand your response, but it appears I've got it working as it should do.

This performs movement in eight directions, and allows for sliding against the side of the box so you don't stop dead moving diagonally against it!

It's also made the code a bit shorter than before I think... I understand I probably shouldn't have four if statements like that for the movement but I don't know how to optimise that in a way that lets it still work.

1. Box player1, environment;
   final static int NORTH = 1;
   final static int EAST = 2;
   final static int SOUTH = 4;
   final static int WEST = 8;
   int result;
   
   void setup()
   {
     player1 = new Box(20, 20, 50, 50);
     environment = new Box(200, 200, 50, 50);
     size(300,300);
   }
   
   void draw()
   {
     background(0);
     player1.move2();
     player1.drawBox();
     environment.drawBox();
   }
   
   int isColliding(Box box1, Box box2)
   {
     if (!((box1.posX + box1.sizeX) >= box2.posX)) return 0;
     if (!(box1.posX <= (box2.posX + box2.sizeX))) return 0;
     if (!((box1.posY - box1.sizeY) <= box2.posY)) return 0;
     if (!(box1.posY >= (box2.posY - box2.sizeY))) return 0;
     return 1;
   }
   
   int inLevel(Box box)
   {
    
    
     return 1;
   }
   
   void keyPressed(){
     switch(key) {
       case('w'):case('W'):result |=NORTH;break;
       case('d'):case('D'):result |=EAST;break;
       case('s'):case('S'):result |=SOUTH;break;
       case('a'):case('A'):result |=WEST;break;
     }
   }
    
   void keyReleased(){ 
     switch(key) {
       case('w'):case('W'):result ^=NORTH;break;
       case('d'):case('D'):result ^=EAST;break;
       case('s'):case('S'):result ^=SOUTH;break;
       case('a'):case('A'):result ^=WEST;break;
     }
   }

1. class Box
   {
     int posX, posY, sizeX, sizeY;
    
     Box(int x1, int y1, int x2, int y2)
     {
       posX = x1;
       posY = y1;
       sizeX = x2;
       sizeY = y2;
     }
    
     void drawBox()
     {
       fill(255);
       rect(posX, posY, sizeX, sizeY);
     }
    
     void move2()
     {
       if ((result & NORTH) == 1)
       {
         posY--;
         if (isColliding(player1, environment) == 1) posY++;
       }
       if ((result & EAST) == 2)
       {
         posX++;
         if (isColliding(player1, environment) == 1) posX--;
       }
       if ((result & SOUTH) == 4)
       {
         posY++;
         if (isColliding(player1, environment) == 1) posY--;
       }
       if ((result & WEST) == 8)
       {
         posX--;
         if (isColliding(player1, environment) == 1) posX++;
       }
     }
   }

To make it a lil' shorter, instead of switch(key), use switch(keyCode).
This way, you only need to test for capital letters:  

case 'W': case UP: result ^= NORTH; break;