Disable Start menu new Items highlighting by GPO in Windows 2012 R2

Unfortunately I don't believe there is a GPO option to disable this. However this option is set by a registry setting so you could create a startup script to set this registry key, to disable this option.

The registry value you're looking for is:

[HKEY_CURRENT_USER\Software\Microsoft\Windows\CurrentVersion\Explorer\Advanced]
"Start_NotifyNewApps"=dword:{00000000|00000001}

It's a REG_DWORD value and should be set to 0 to disable the highlighting, 1 to enable it.

One alternative would be to create a rule in Widows Firewall that blocks outbound traffic to the port ranges (or even from the actual program creating the traffic) you are seeing in group policy. That would take the load off the network devices.

Create an Outbound Port Rule on Windows 7, Windows Vista, Windows Server 2008 or Windows Server 2008 R2

This may/should also result in DropBox itself disabling LAN Sync (based on the DropBox LAN Sync help page):

If Dropbox detects a firewall preventing access to your LAN, it will turn off LAN sync in your Dropbox preferences automatically.