Couple things is

1. Verizon does not run fiber inside the house they use standard RG-6 coaxial cable the fiber is only to the outside where there is a network interface that converts it over to coax.

2. Unlike the benefit of having cox  with verizon you need to have a set top box in order to watch tv. At my house i have cox and have one hi def dvr box, but i also have 6 cable ready tv's. If i were to switch to verizon i would need a total of 7 set top boxes which they charge for the rental. It would be a ridiculously high bill.

3. so to answer you will be able to run your own outlet using coaxial wire but you will still need verizon to come out and install the set top box.

4. Biggest reason Im staying with Cox. Verizon deceives you into thinking they bring fiber directly to your tv or computer. Completely false. Cox has fiber optic cable in place also but they dont convert it at the house its converted generally within 1/4 mile of any customers home.