Assign null, I've just done this:

var a=1;
console.log(a);
a=null;
console.log(a);
console.log(a===null);

Results:

1
null
true

Though, null in JS is also a value.

See the specification:

1. Let JText be ToString(text).

It doesn't require that the first argument be a string, it attempts to convert it to a string.

"" + null will give you "null" which is a string containing valid JSON.

This is also in the spec:

Null: Return "null".

A null check in JavaScript if just like any other C-style language:

 if (thing == null) 

Or

 if (thing != null)

I find this works well in most cases against my own programming where I'm writing as I would in, say, C#; however I find other peoples code relies on things never having been declared or set and such and so, and, all in all, it boils down to a spaghetti of checking for null and "undefined" - yes, the literal string, really - and whatever else.

I just want something that would evaluate to false

You could use an empty string instead to mark the unset value:

sessionStorage.setItem('key','');

This way you can re-use your current check:

if (!sessionStorage.getItem('key')) {
   // do something
}

That said, you mentioned you had a drop down to select values, and that's why you need such a value, but I think a solution involving deleting the item on the null value would be better. Here's a working demo, and the code used to make it:

var dd = document.getElementById('dropdown');

dd.addEventListener('change', function(){
  var value = this.value;
  if (value === 'null')
    sessionStorage.removeItem('key');
  else sessionStorage.setItem('key', value);
});

var check = document.getElementById('check');
check.addEventListener('click', function(){
    this.innerHTML = 'Value: '+sessionStorage.getItem('key')+
                     ', check: '+(!sessionStorage.getItem('key'));
});

<select id="dropdown">
  <option value="null">none</option>
  <option>1</option>
  <option>2</option>
  <option>3</option>
  <option>4</option>
  <option>5</option>
</select>

<button id="check">Check value</button>

How about:

var convertType = function (value){
  var values = {undefined: undefined, null: null, true: true, false: false}
     ,isNumber = !isNaN(+(value));
  return isNumber && +(value) || !(value in values) && value || values[value];
};
convertType('null');      //=> null
convertType('something'); //=> "something"
convertType('57.321');    //=> 57.321
convertType('undefined'); //=> undefined

This seems faster @ jsPerf

var convertType = function (value){
    var v = Number (value);
    return !isNaN(v) ? v : 
         value === "undefined" ? undefined
       : value === "null" ? null
       : value === "true" ? true
       : value === "false" ? false
       : value
 }

Your code is correct check out this JSFiddle.

var JSONResponseFromServerSide = '{"command":"SELECT","rowCount":1,"oid":null,"rows":[{"username":"xxxx"}],"fields":[{"name":"username","tableID":34722,"columnID":3,"dataTypeID":1043,"dataTypeSize":-1,"dataTypeModifier":204,"format":"text"}],"_parsers":[null]}'; //it's your json string
$.parseJSON(JSONResponseFromServerSide); //nothing wrong

If JSONResponseFromServerSide is already a JavaScript object, than you don't have to do parseJSON

You can also use the isNaN() function:

var s = ''
var num = isNaN(parseInt(s)) ? 0 : parseInt(s)

Ignoring whitespace strings, you could use this to check for null, empty and undefined:

var obj = {};
(!!obj.str) // Returns false

obj.str = "";
(!!obj.str) // Returns false

obj.str = null;
(!!obj.str) // Returns false

It is concise and it works for undefined properties, although it's not the most readable.