You can add caller ID blocking to your line through your MyVerizon account online.
May have previously been available in myverizon but is no longer.
Searched 30 minutes before posting question.
Can only turn caller-id on/off for incoming calls under 'Change Features'.
Verizon-CDMA iPhone 5 does not have this as user-configurable setting.
Contact VZW to see if they can do it on their end (I think they can).
You were right -- they can and did. This is not documented (properly or well, depending on where you look and what phone you are using in your search criteria; I have an iPhone 5). There are drawbacks to the blocked call ID but for my need (separating work and personal calls with a privacy "wall") this was the solution needed.
Finally, I was surprised how helpful and willing the live-support (#611) people were -- compared to the staff at Verizon stores. Poor folks in the stores are only trained to open accounts and sell phones, with virtually no knowledge of basic telecom services or features. This frustrated me like crazy because I made the mistake that at least one person in the two stores I went to would know what caller id was or had seen an iPhone 5 before.
Anyway, thanks for your reply. Just affirming that it was the solution I needed.
You said its a new account but I wanted you to be aware anyway if you are not on the Share everything plan and on the 700 minute, or if somehow you did get a nationwide plan:
When your caller id is blocked, the mobile to mobile and frieds/family feature does not apply unless you unblock your number prior to the call.