http://forum.processing.org/topic/simple-json-example-with-api

The Json you show is full of errors...
I was able to parse this version:

1. String input = "{\"type\":\"grid\"," +
2. "\"grid\":" +
3. "[" +
4. "[0,0,0,0,0]," +
5. "[0,0,0,{\"x\":1,\"y\":3},0]," +
6. "[0,0,{\"x\":2,\"y\":2},0,0]," +
7. "[{\"x\":3,\"y\":0},{\"x\":3,\"y\":1},0,0,0]," +
8. "[{\"x\":4,\"y\":0},0,0,0,0]" +
9. "]}";
10. JSONObject json = parseJSONObject(input);
11. println(json);
12. exit();

Now, it cannot go into a "Processing data structure", because it is made of heterogeneous data, mixing integer values (zeros) with objects (x, y). Unless you replace the 0 with null.
Otherwise, you have to iterate on the Json data structure and extract information on the fly.

I was able to extract some information from your Json data:

1. import processing.data.*;
2. void setup()
3. {
4.   //String input = "{ \"data\" = [ 5, 7, 9, 11 ] }";
5.   String input = "{\"type\":\"grid\"," +
6.     "\"grid\":" +
7.     "[" +
8.     "[0,0,0,0,0]," +
9.     "[0,0,0,{\"x\":1,\"y\":3},0]," +
10.     "[0,0,{\"x\":2,\"y\":2},0,0]," +
11.     "[{\"x\":3,\"y\":0},{\"x\":3,\"y\":1},0,0,0]," +
12.     "[{\"x\":4,\"y\":0},0,0,0,0]" +
13.     "]}";
14.   JSONObject json = parseJSONObject(input);
15.   //println(json);
16.   String type = json.getString("type");
17.   JSONArray array = json.getJSONArray(type);
18.   //println(array);
19.   for (int i = 0; i < array.size(); i++)
20.   {
21.     JSONArray a = array.getJSONArray(i);
22.     println(i + " " + a);
23.     for (int j = 0; j < a.size(); j++)
24.     {
25.       try
26.       {
27.         int d = a.getInt(j);
28.         println(j + " " + d);
29.       }
30.       catch (RuntimeException e)
31.       {
32.         JSONObject d = a.getJSONObject(j);
33.         println(j + " " + d);
34.       }
35.     }
36.   }
37.   exit();
38. }

I don't like this try / catch business, but the API currently doesn't offer a way to find out what the kind of data is at a given index.
Now, Json is better used with regular data where the type at a given place is of expected type...

Thank you very much for the example, PhiLho.  That JSON was actually produced by a node.js server using JSON.stringify(), so I'm surprised Processing's libraries can't read it.  The critical difference appears to be that your JSON array isn't surrounded by quotes whereas the one produced by the server's is. 

" your JSON array isn't surrounded by quotes whereas the one produced by the server's is"
Yes, that was one of the errors, and the fact some quotes were escaped while others were not...

" I'm surprised Processing's libraries can't read it"
Beside the errors, JavaScript is much more flexible than Java in its data structures, so a 1-1 mapping isn't possible.
In Java, it is nearly impossible to make a variable able to store either an integer or a pair (or more) of coordinates. Unless making a special class, in a quite convoluted way.