After enabling Selective Sync for Windows or OS X agents they register context menu extensions in the system. These allow agents to show context menu items.
Before troubleshooting double check that Selective Sync is indeed enabled for the agent or its group in a job.
OS X
1) Try resetting Finder Extension.
Go to System Preferences > Extensions > Clear and then reselect the checkbox for Sync. Uncheck Resilio Connect Agent and check again. Relaunch Finder: press Alt + right click on Finder in Dock -> Relaunch or click on Apple -> Force Quit -> Finder.
2) If that didn't help, uncheck other apps' extensions and again relaunch Finder, restart the agent.
If neither tip helped, Contact support and submit debug logs from the agent. Also, collect system.log file and send that to tech support as well. To collect system.log:
- open Console.app
- reproduce the problem with context menu
- select system.log in the left pane, select the content, and copy paste it to a text document. 
WINDOWS
Context menu items will show up for shares located on NTFS file system - system that supports alternate data streams.
1) Relaunch Windows Explorer process.
2) Open Elevated Command Prompt (run as admin) and navigate to agent installation directory (C:\Program Files\Resilio Connect Agent by default) and re-register the two dll files there, using command regsvr32 /i
And relaunch Windows Explorer again.
3) If that does not help, open Registry Editor and set key HKEY_LOCAL_MACHINE\SOFTWARE\Microsoft\Windows\CurrentVersion\Policies\System\EnableLUA to 1. Reboot PC.
If their of suggested tips helped, Contact support and submit debug logs from the agent.