If they are DVRs, they turn on to record and turn off again after recording is done.
I have also accidentally hit remote buttons on the remote.
I have Comcast (no FiOS here) but they use the same Motorola boxes that Verizon uses.
If you really don't like Verizon invading your home, you can always go the TiVo route as Verizon's signal ends at the cable card and they are one-way devices so they can only receive downstream signals but they can't talk back to Verizon on the upstream side. TiVo boxes don't turn off but they are a little more user friendly than the Moto boxes. -- I wish I still lived in Iowa; Everything there from rent and groceries to Cable TV is much cheaper in Iowa (especially with an overbuilder in town).