Yes, whenever you upgrade the firmware on your child's or employee's iPhone, it will remove all programs and utilities associated with jailbroken software. This will include Mobile Spy and Cydia.
You should not upgrade firmware until you have time to re-jailbreak your device and make sure that firmware version can be jailbroken. Apple, lately, has been working on creating firmware versions that cannot be jailbroken. Make sure if you plan to upgrade that a reliable jailbreaking method for that firmware is available.
So if you upgrade the iPhone's firmware and it can be jailbroken, simply re-jailbreak the device and reinstall the software through Cydia.