This should do it:
item.innerHTML += "<li><input type='checkbox'>" + document.form.itemEntered.value + "</li>";
The issue you experienced is called escaping.
You are generating a string in JS using double-quotes so if you want to use a literal double-quote inside your string then you need to escape it, please study the code below:
item.innerHTML += "<li><input type=\"checkbox\">" + document.form.itemEntered.value + "</li>"; //good
item.innerHTML += '<li><input type="checkbox">' + document.form.itemEntered.value + '</li>'; //good
item.innerHTML += '<li><input type=\'checkbox\'>' + document.form.itemEntered.value + '</li>'; // good
item.innerHTML += "<li><input type="checkbox">" + document.form.itemEntered.value + "</li>"; //bad
item.innerHTML += '<li><input type='checkbox'>' + document.form.itemEntered.value + '</li>'; //bad
The devil is always in the details.