Things it's not
First off, busybox isn't licensed under GPLv3.*
Nor would I call it a 'network utility' even if some of the things it has in it are network utillities.
Apart from that, yeah, if Verizon added anything to it, then they'll need to pony up the source for what they added. Seems unlikely that using it 'infects' their whole box though, but the lawyers will decide that.
* I didn't check whether any of the things in busybox are GPLv3, but that too seems unlikely as doing so would have infected busybox.