I always try to use what the cable company, or in this case Verizon uses. Personally I don't think dual vs quad will make a difference either but if Verizon is willing to pay for quad there must be a reason. First, make sure your information comes from Verizon regarding quad being necessary, not someones opinion. Then, if you ask nicely most cable companies will give you a box of their cable and I can't believe Verizon won't do the same. After all, you are doing them a favor by pre-wiring for their service.
I would let them terminate though unless you want to see them cut your fittings off and replace them with theirs.
-Hal