First, you don't need form tags to do this. That will force the page to reload every time it completes, thereby erasing everything, including your results. Change the form tags to div and it should be fine.
This is not a legal conditional expression:
if (!rectangleLengthInput.value || !rectangleWidthInput.value == none) {
A proper OR expression testing two conditions would be:
if (!rectangleLengthInput.value == none || !rectangleWidthInput.value == none) {
But, of course, there's a problem. "none" does not exist as a defined value in javascript. To check for no value in a textbox, you can do something like this:
if (rectangleLengthInput.value === "" || rectangleWidthInput.value === "") {
Moving on...Your calculations are also wrong. You're trying to do calculations using the textboxes themselves, not the values in the textboxes. I leave it to you to figure out why.