I have a crude way of changing the login screen but it does work without any problems.
Go to file system/usr/share/backgrounds and open the folder linux mint as admin/root. Right click on "default_background" and open it in gimp. (assuming you have gimp or something similar)
Open the background image of your choice in gimp as well. Click edit and cut to remove the image out of the "default_background". Copy the image from your favorite background image and paste into the default background file. Resize the image with the resize tool till it fits and then crop so it fits. Save the image, which should work if you have permission to edit the file.
Restart linux and you should see the background of your choice as your login screen... Like I said, the method is crude but it works well if done properly