In the Gospels, Jesus speaks about thirst a few times. When he does, he doesn't tell his audience to stay thirsty. In fact, he says the opposite. In Matthew 5:6, Jesus states that those who hunger and thirst for righteousness will be filled. In John's gospel, Jesus invites those who are thirsty to come to him and never thirst again (John 7:37 and 6:35). In John 4:14 he says that those who come to him will not only never thirst again, but that this living water will pour out from them. Doesn't sound like Jesus expects those who come to him to stay thirsty.
I believe those who put the sign up meant well. There is a sense in which we need to remember that only Jesus can quench our thirst, because there are times we forget and look for other things to satisfy us. Maybe they meant that we must keep coming to Jesus, and not think that a one time experience is enough. If that's the case, I can agree with that.
Although I still cringe at churches that rip off culture on their signs. :)