Firstly lets combine some knowledge:
How do I test for an empty JavaScript object?
The above question has an awesome answer with many options in it, I'm going to edit the snippets to form a couple of functions:
function isObject (obj) {
    return obj.constructor === Object;
}
function isEmptyObject (obj) {    
    return Object.keys(obj).length === 0 && isObject(obj);
}
Ace. Now, assuming you have a very linear object structure, it's not a complex array and don't care about the depth (only that it is empty) lets make a function loop through the tree.
function convertEmptyObjects (obj, currentDepth) {
    currentDepth = currentDepth || 0;
    for (var key in obj) {
        var val = obj[key];
        if (isObject(val)) {
            if (isEmptyObject(val)) {
                obj[key] = "{}";
                console.log("Defeated boss ("+ key +") on level "+ currentDepth +"!");
            }else{
                convertDeepestEmptyObject (val, currentDepth + 1);                    
            }
        }
    }
    return obj;
}
Lets test this on an awful looking object:
var testObject = {
    CONN_INFO: {
        CFGSwitch: {
            412: {}, // Level 2
            413: {
                content: {}  // Level 3
            }
        },
        DummySwitch: {} // Level 1
    },
    TEST_CONN_INFO: {
        CFGSwitch: {
            414: {},  // Level 2
            415: {
                content: {
                    host: "google",
                    port: "8080",
                    protocol: {}  // Level 4
                }
            }
        }
    },
    STAGING_CONN_INFO: {} // Level 0
}
convertEmptyObjects(testObject);
JSON.stringify(testObject)
/* Output:
* Defeated boss (412) on level 2!
* Defeated boss (content) on level 3!
* Defeated boss (DummySwitch) on level 1!
* Defeated boss (414) on level 2!
* Defeated boss (protocol) on level 4!
* Defeated boss (STAGING_CONN_INFO) on level 0!
*/
// Result:
{
    "CONN_INFO": {
        "CFGSwitch": {
            "412": "{}", // Empty String {}
            "413": {
                "content": "{}" // Empty String {}
            }
        },
        "DummySwitch": "{}" // Empty String {}
    },
    "TEST_CONN_INFO": {
        "CFGSwitch": {
            "414": "{}", // Empty String {}
            "415": {
                "content": {
                    "host": "google",
                    "port": "8080",
                    "protocol": "{}" // Empty String {}
                }
            }
        }
    },
    "STAGING_CONN_INFO": "{}" // Empty String {}
}
There you have it.