I am not an expert by any means, but I was having the same problem. I have a samsung phone, and somehow this update app was installed on my mac (I probably did it myself without realizing it).
But I do not update my phone through my mac, so this is unnecessary and kind of annoying. So, I kind of stumbled around to figure out how to remove it. Here's what I found.
Open up the search spotlight, and type in "Verizon Software Upgrade Assistant." You'll get a window that shows the Verizon Wireless programs which are on your mac. Under Folders, click on Verizon Software Update. That will give you something like this:
Simply move each of these items to the Trash. One of them will probably not move, because it is open. Just click on that one in the task bar or wherever its icon is, and close it, then it will move. After you have moved each of these to the trash, simply close that entire folder, and try logging in again. The Verizon Software Update reminder should be gone.
I do not know if this actually uninstalls the program. Perhaps there are persons more knowledgeable who can improve on what I did.