This article was published in 2014. Is Verizon still doing this to customer data? It makes sense to assume they are, but I'm hoping for some kind of official news article or something to validate its continued existence.
Despite a 2016 FTC settlement against an advertiser abusing the X-UIDH header and a 2016 FCC settlement (PDF) with Verizon, Verizon's legal pages indicate that it is still in use in some circumstances.
There have been changes to FCC rules within the past couple of years, impact on this issue unknown.
There are some user controls in the form of opt-outs and opt-ins to various Verizon advertising preferences that affect the X-UIDH header:
Verizon Wireless stops inserting the UIDH after you opt out of the Relevant Mobile Advertising program, unless you have also opted in to the Selects advertising program. The UIDH will continue to appear for a short period of time while we are updating our systems. If you choose to participate in Verizon Selects, the UIDH will be present even if you have also opted-out of the Relevant Mobile Advertising program.
It's only possible for Verizon to inject the header on HTTP traffic. HTTPS (try EFF's HTTPS Everywhere browser add-on to use as much HTTPS as possible), or Tor, or a VPN, will block this behavior.
You can check for the X-UIDH header on any unencrypted (HTTP; not HTTPS) website that displays all of your HTTP request headers.