This plan. I like this plan! Put a random value in the header on every request. If you're not on Verizon, it'll look like you are (but as a different person every time). If you *are* on Verizon, you may just confuse the software that is adding those headers, or that is logging them. Poison their tracking data with meaningless garbage, and make it *cost* Verizon money to try and track us.
Well, that and use HTTPS everywhere possible, of course. But that requires that the sites you use allow people to do so (*AHEM* Slashdot, looking at you...)
Oh, and don't use Verizon. That's the best way to hit them in the pocketbook, by far. I like the idea of sending the header even when you don't use Verizon though, as a general-purpose "fuck you!" to them.