I just downloaded Gog Galaxy 1.1.28.49. When I start it says it can't update, and when I'm inside it won't Connect. I turned off my firewall completely and it still won't connect. I'm running Windows 10 Home with the latest updates.

Hi,
I also had this kind of problem, in my case the Client couldn't update from version 1.2.17 to 1.2.20.
The solution for me was to uninstall and reinstall the Galaxy client (run as administrator).
After starting the Client app the updater tried again, multiple times, to update and failed.
But when I closed the App and started it again, the updater finished quickly and showed an higher version than the last time. So it finally worked!

I'd found out my games were "gone", from the left bar.
This was because I renamed the Installation folder of the Client with an "Games" folder in it.
So I moved the Games folder into the new folder. After that I clicked the Game title in the library, and clicked "Manage Installation" to import the current Games folder.
After that the files where verified and updated automatically.
The Game Library filled again with my Game titles, and after launching it the saved progress was still there.

For other users with this kind of problem it may be better to uninstall, download the latest Galaxy version from gog.com and install that version. For the rest the next steps might be useful.