Am I able to detect if a phone has been jailbroken if it is managed/supervised by Apple Profile Manager?
Profile Manager isn't an MDM that reports details about app install which is usually used to determine jailbreak status. (i.e. If cydia app and other apps are installed, it's a good conclusion that the device has been jailbroken.)
Also, once jailbroken, it's entirely possible the device won't report back the correct information depending on what else is modified past the initial compromise of device security.