How to fix broken nautilus-dropbox icon in ubuntu 18.04 GNOME-Flashback in top Panel?

To sum up:

1. Run

   dropbox stop && env XDG_CURRENT_DESKTOP=Unity dropbox start
   

2. Click on Dropbox icon after it starts up. Go to the "Preferences" and uncheck "Start Dropbox on system startup" (to prevent dropbox from changing the startup command after reboot.)

3. Go to Applications > Preferences > Startup Applications, if Dropbox is already there click on Edit, and if not click on Add.

   Enter
   Name: Dropbox
   Command: env XDG_CURRENT_DESKTOP=Unity dropbox start
   Comment: Sync your files across computers and to the web

4. Run

   gedit ~/.config/autostart/dropbox.desktop
   

   replace Exec=dropbox start -i with

   Exec=env XDG_CURRENT_DESKTOP=Unity dropbox start
   

and

5. Run

   sudo gedit /usr/share/applications/dropbox.desktop
   

   replace Exec=dropbox start -i with

   Exec=env XDG_CURRENT_DESKTOP=Unity dropbox start
   

I can confirm it on clean Ubuntu 18.04 LTS install with GNOME FlashBack session, so I installed only GNOME Panel new package and Nautilus Dropbox plug-in

sudo apt-get install gnome-panel nautilus-dropbox

After installing of nautilus-dropbox I have logged-in to my Dropbox account.

Then after rebooting system I get the same image as OP, so I reported bug 1767665.

It may be fixed locally as @Alberts Muktupāvels (muktupavels) suggested on launchpad.
I shorten method to just two simple lines:

mkdir -p ~/.config/autostart/
cp  /usr/share/applications/dropbox.desktop ~/.config/autostart/dropbox.desktop
sed -i "s/dropbox start -i/env XDG_CURRENT_DESKTOP=Unity dropbox start -i/" ~/.config/autostart/dropbox.desktop

The first finds Exec line dropbox start -i in it and changes it to env XDG_CURRENT_DESKTOP=Unity dropbox start (it forces dropbox to "use" Unity desktop even if we are using GNOME), the second creates user XDG-autostart directory, the third line copies XDG-autostart desktop-file to user XDG-autostart directory.

Afterwards Dropbox menu is in place and works as expected.