Can I us an External Hard Disk on two Computers?

Yes, any External drive can be used on various systems and be accessed. you can put as many files in as many folders as you want. Until the drive is full, of course. And it's your own responsibility that files from comp A don't overwrite files from comp B by putting them in different folders.

You can install a program on an external hard disk if the setup allows you. It's up to them to do it or not. But if it relies on things the install puts in the registry, you'll need to do the install on both comps. And things might go wrong if this drive gets different drive letters on both comps. So it's somewhat tricky. I've got certain programs I can run from a USB-stick, and if that works it should work from an external hard disk also. All I can say is try it, but don't trust on it until you know it works.

Yes, you can use this facility with using it as a "backup" HD will cause me to reconsider as you plan to use on different systems for the same purpose, backup. After all, a backup is critical to one system to place that much demand for two systems can place them in dire need should the ext. HD fail. If you review here all the issues many an ext. HD experienced, don't be so dependent on the ext. HD for super critical backup. As Keeps explained be sure its noted the folders (or partitions) are clearly noted for each system should you proceed. If possible, get yet another ext. HD to support one per system as a better solution.