You can click screenshots within your Mac but unfortunately you can't take screenshot of the login screen. There are quite a bunch of apps that can probably get it done but what if you don't want to resort to apps?
You can take a screenshot of the login screen but you'll need two Macs to configure this setting. We're going with two Macs here (a Linux distro will do just fine but you'll need a screenshot library like the gnome-screenshot or nanoshot). Note that both the Macs/computers need to be on the same network for this to work.
How to Take a Screenshot of the Login Screen on Mac
On your first Mac, you need to do this:
Step #1. Click on “Apple” menu and then select on “System Preferences”
Step #2. Click on “Sharing.”
Step #3. Click on “Lock” icon at bottom (so that you can make changes)
Step #4. On the left pane, look for a checkbox labeled “Remote Login.” Check it.
Step #5. On the right pane, under Remote Login, there will be text saying, To Log in to this computer remotely, type “ssh….”
Step #6. You need to save the text “ssh….” as you need to type it in your second Mac. Copy it on your second computer/Mac.
Step #7. Logout from your Mac and return to the Login screen
On your second Mac, do this:
Step #1. Select “Applications” > “Utilities” > “Terminal.”
Step #2. Write the “ssh…” command, you noted down from your first Mac and hit return (Enter).
Step #3. Your second Mac will remotely login into your first Mac. You can confirm this by the bash/terminal command line. It would have changed to the name of the first Mac.
Step #4. Type: cd ~/Desktop and press the return key.
Step #5. Now, type: sudo screencapture -tjpeg loginwindow.jpeg and hit return again.
The two commands in 4 & 5 will basically allow your second Mac to enter into your first Mac's desktop and save the screenshot with the name: “loginwindow.jpeg.” So, when you login to your first Mac, you will see the screenshot saved on your desktop. That's about it.