Yes, game files can be transferred between OSs with no problem.
The easiest method is to use the import/export game feature. From the File menu, choose Export Game... and save it to the USB drive, and then at home or school, the .stencyl file can be dragged right in to Stencyl, or you can use the Import Game... option.
Another method is to do the work yourself with a file explorer. You can open your file explorer to the location where games are saved using the menu bar: Debug > View > View Games Folder. It's also possible to change where games are saved on the computer using the workspace setting: File > Preferences > Workspace > Workspace Folder
Finally, you could also use StencylForge to upload a game for private storage.