I have found Verizon to be hit and miss in Mexico. I have Verizon, and in December 2012 and December 2013, I set up the Mexico Roaming Plan with Verizon before heading off to Cozumel, and the phone (an iPhone, 4 years old now) worked just great. Then I did the same thing in preparation for my December 2014 trip to Cozumel, and I couldn't get my phone to work for 3 days - I got in touch with Verizon through the Website, Live Chat, and we went back and forth trying to fix it, finally got the phone to work on the 4th day. Verizon tried to blame the local cell carrier for not recognizing my phone, but couldn't explain why it worked without a hitch the previous 2 years.
---------- Post added June 6th, 2015 at 07:52 PM ----------
Yes, DEFINITELY turn off data on the phone. Verizon will RAPE you with data charges if you don't. You can do it on the iPhone through Cellular - Settings. If you don't turn it off, and your iPhone is set to check for and download email, you will be very unpleasantly surprised when the bill comes. It will be even worse if you use the GPS function to find locations, or, heaven forbid, surf the web on your iPhone while in Cozumel.