is there a program that will...

[dynamite boy]

...download everything from a single website directory that isn't linked from the site itself but doesn't have a full file listing?

 

[BernOrange]

an FTP client?

 

[blacklab]

curl
kinda does that if you know what the file names are like.

a perl script could be written to cycle through and grab all the files pretty easily. (might take a while to run though and your ip will probably get booted from the server)

 

[hellbunny]

If you can't login to the server and
if the directories have basic listing protection and
if the files aren't all linked from somewhere and
if you can't guess the file names... then no.