I think the key phrase here is [and then plug it into my new computer]
No, it wont work at all, because the operating system has to match the Motherboard it was installed on, and most likely will not even boot.
By all means partition your new external HD, though recognisable new folders are all you really need.
You can use your new HD to store an image of your OS, in fact this is one way of having extra safety if the image was made using Acronis or latest Ghost software. The idea is that when your OS goes belly up you can restore a working system to the original drive using the Software method.
You can of course use the new HD as a vast place for storing every sort of data, dont bother with partitions just make suitable folders and sub folders as necessary.