I noticed a few other posts with Verizon Messages troubles in the past few weeks, but tonight it looks to be straight unavailable.
I tried connecting through Chrome (the page itself times out) and IE (the Verizon page loads but the web app itself just has a spinning wheel, forever) on two different computers (a Windows 7 desktop and a Windows 8.1 tablet) and my cell phone (Windows Phone 8.1). My internet connection is Verizon Wireless 4G LTE, so it would be a bit bizarre if it were an ISP issue; the rest of Verizon's site, including My Verizon, all loads just fine.
I have used it in both browsers, on multiple devices, as recently as yesterday without any issue. I tried clearing all browsing data (cache, passwords, security certificates, etc.) and restarting but I still get the same thing. All other internet traffic is, obviously, coming through just fine.