iMessage needs Internet connection on both ends (cellular network or through Wi-Fi)
Well, according to what you wrote Verizon network may be the problem.
My daughter is also using the iPhone 4S with IOS 7.0.6 and she's texting like mad which I have to stop her using it after 9pm.
You can make an appointment with the Apple genius bar for an evaluation.
In the meantime, you can check whether the Apple iMessage server is down.