How do I uninstall the Epson Printer Update App?

You can check to see if you've removed all of the supporting files by downloading and running the shareware app Find Any File to search for any files with the application's or the developer's name in the file name.  For Epson software you'd do the following search(es): 

1 - Name contains epson

2 - Name contains "name of app you deleted"

Any files that are found can be dragged from the search results window to the Desktop or Trash bin in the Dock for deletion.

FAF can search areas that Spotlight can't like invisible folders, system folders and packages.  

If you get warnings that the file can't be deleted because it is in use or used by another app boot into Safe Mode according to How to use safe mode on your Mac and delete from there.

Note:  if you have a wireless keyboard with rechargeable batteries connect it with its charging cable before booting into Safe Mode.  This makes it act as a wired keyboard as will assure a successful boot into Safe Mode.

How did you uninstall the printer files in the first place? Did you just move them to the trash? Usually, such apps have a dedicated uninstaller that gets installed along with the printer files.

Please check your Applications folder for such an uninstaller (strangely, in some cases you might even need to install the app again (so the uninstaller will be downloaded as well) and then you can use it to finally uninstall the app correctly along with all the leftovers)

You can also try to remove associated files manually:

• Open Finder and press Command + Shift + G.
• Enter ~/Library/ and click Go.
• Search for and delete any files or folders related to Epson in the following directories:
  • Application Support
  • Caches
  • Preferences
• Repeat the process in /Library/ (without the tilde ~) for system-wide files.

You can also consider using a third-party uninstaller since they remove the app and all its associated files (contrary to just moving the icon to trash)

I also agree with the rest of the answers that you should boot in Safe Mode first and only after that remove the files

Have you asked Epson themselves how to stop the popup?

Most likely it is located in the Applications folder. Locate it and drag it to the Trash and empty it.

If a file(s) is in /Library can you still delete it using Safe Mode?