If you already broke the screen, then insurance wouldn't help since you didnt have insurance when it broke. The deductible would be quite high even if you had ins. Replacing the screen locally will be much cheaper. But, be careful, if you get insurance after the fact and need to file a claim, the repair facility would know it had been repaired before. Most ins providers replace with a refurb anyway. You might want to check around online. Most will insure within 30 days of purchase of a new phone.
Apple will replace the screen for you. That is normally less expensive than say a deductible. Snn5 is correct that since the device is already broken insurance can't really help in your situation.
Here is Apple's site for more information. iPhone Screen Damage Repair - Apple Support
If you go to a 3rd party retailer they could be using non factory (OEM) products thus it is good to have Apple replace it.. especially if there is an Apple Store close to you.
Being able to rely on your phone is a big deal. I want to make sure you can always count on getting it fixed if you need to. If you get insurance through Verizon it does have a deductible if it needs to be fixed or replaced. That would be taken care of through our insurance partner Asurion, not through Apple.
If you go through Apple, they will be glad to go over any options they have. If there is any cost that would be up to them but it wouldn't be tied to the insurance in any way. Is your phone having any issues? We're glad to help.
Follow us on Twitter @VZWSupport
If my response answered your question please click the �Correct Answer� button under my response. This ensures others can benefit from our conversation. Thanks in advance for your help with this!!