If I am wiring a garage and the Verizon people want me to put an outlet in for "them"; and them only, is that dedicated circuit required to be GFI protected?

Yes it is.

Good question.

All outlets in a garage need to be GFI protected, however,

If the Verizon system can be related to a fire or burglar alarm system, or if the phone system is not directly related to Life Safety, and a GFI is required, check with the local inspector otherwise to see if this can be waived. I would not want my phone system disabled by a tripping GFI.