There should be an auto-save page with 3 slots on the loading screen. You can reach it by navigating back from Page 1 and forward from Page 2 of the Load Menu.
If those slots are empty after playing for a while, I suspect you may be on a Mac. Be sure to extract Henchman Story from the DMG to your Applications folder before playing, or saves won't work. The DMG is provided so the app can be certified as safe by Apple, sorry if that's an incovenience.
Hope that helps, and thanks for supporting the game!
The regular save files should be working, but you have to save in them manually. If you've unzipped the files to a folder you have permissions to add or alter files in, there shouldn't be any issues with saving. If you continue to have issues, could you describe the exact process you go through when saving and where the problem occurs? You should just be able to open the Save Menu (not the Load Menu, which looks similar) and click on one of the save slots, which look like comic books.