Will Mobicip work in Incognito/Private mode?

Yes, Mobicip does filter, monitor and record activities done through any browser and in private browsing mode/incognito mode too. 

You just need to perform an extra step in case of Chromebooks - To allow Mobicip in Chrome Incognito mode, click on the puzzle icon to list extensions > Click Mobicip > Uncheck 'Block extensions page' > Login with account info > Here you can enable 'Allow in Incognito' in the list of settings available.