Can the JS array length property ever return a negative value?

No it cannot. MDN says:.

The value of the length property is an integer with a positive sign and a value less than 2 to the 32nd power (232).

No.

The spec for the length property says:

The length property of this Array object is a data property whose value is always numerically greater than the name of every deletable property whose name is an array index.

There cannot be -1 properties.

Also, and more explicitly, the spec for Array says:

Every Array object has a length property whose value is always a nonnegative integer less than 2³².

Update: Answer still holds for ES2020

Not normally (as other answers have pointed out), but an object that is an instance of an array can have a negative property called length

var b = Object.create([]);
b.length = -1;

alert(b instanceof Array)
alert(b.length);

No, it can't. Moreover, if you try doing this arr.length = -1 it will throw you an exception