I personally don't know if Verizon's offering will help you. There are "spy" apps out there that can supposedly track this method of messaging, but I'm not sure if they can block it. You could Google Whatsapp Parental Controls.
If your child has downloaded and installed Whatsapp - then likely there is a trust issue and they are trying to circumvent SMS/MMS texting restrictions or monitoring...
When they find out that Whatsapp is being monitored or restricted, they'll likely move onto something else.