In case anyone (like me one year into the future when I've forgotten this) is looking for a simple installation guide (what to do with those .sh files), here's what I did:

System: 32-bit Ubuntu MATE, ASUS eeePC (old netbook)

I renamed the jillblablabla.sh file to jill.sh and plopped it in my GOG folder. I couldn't just left or right click on the .sh file though. Instead I had to use "File/Open in Terminal" (i.e. open a terminal set to the current folder) from the window/file manager thingy then run the .sh from the terminal using:

bash jill.sh

It installed, but generated an error that seemed inconsequential. MoO did the same I remember.

However, when I tried to run Jill from the menu link, DOSBox would just exit right away. The terminal also blinks away (can't read the error message if any) so I had to run it from a persistent terminal window (again opening a terminal from the file manager, then typing bash start.sh). The error? Turns out the resolution on my eeePC monitor is too low (1024x600 or something). I went into one of jill's .conf files and set fullscreen=false and that fixed it. That was my particular problem, but if DOSBox just shuts down you might be able to catch & fix other errors using this method.

Linux is kinda crazy about the permissions stuff but I didn't have to fiddle with it in this case (maybe I already had for the GOG folder... who knows, linux is just typing stuff suggested by strangers on the internet then forgetting all about it).

who knows, linux is just typing stuff suggested by strangers on the internet then forgetting all about it

Exactly.

Was having this problem on multiple games and couldn't get them running.

Thanks for the tip!

Thank you for sharing this!