Trying to figure this out was a little bit of a challenge and am in the process of learning about computers. I am trying to find out information on how to install an operating system on an external enclosure and operate a stand alone program off of the external enclosure. I am completely new and had to have my son help me get this far so this is my first shot.

Thanks for any help and patience.

Linux should install onto any disk it sees - including externals.
Booting it is a different matter. Your BIOS needs to support it (e.g. boot from USB), and you may need an initrd. Probably need to delay the mount of the root filesystem via a boot-loader parm (rootdelay) as well.
Note all conjecture on my behalf; haven't bothered trying this.
A boot floppy would also be an option to get the process started.

You'll need to expand on what you mean by "stand alone program" to ensure we are talking the same thing before I could comment on that.

What is the end goal?
Depending on what you are doing, having a an extra internal HD might be much easier.
Also, "external enclosure" is ambiguous. Do you mean external hard drive?

Do you want to be able to plug your HDD into a system and be able to boot YOUR OS?

If so you're going to have trouble with different hardware, so you'd need to make sure the kernel supports everything (maybe compile your own with old legacy drivers etc.) You will also need some scripts to load the correct modules (udev might do this, i keep meaning to look into that).

Like statted above, all the machines will also have to support booting over that interface (USB?) and you will need a floppy disc or something to boot with as all these machiens aren't likely to have bootloaders.

You can have internal enclosures, ie: spuds or hot swap drive caddies.

OK---fine. The main point was to clarify what OP wanted.

Absent any more posts from OP---this thread should now die.