No. It doesn't matter what router you use, but if that wifi icon isn't lit up, you won't be able to surf using wifi. If the router you're using isn't working right, your BB won't connect and you can't use wifi. If you're not connected, the white wifi icon will be greyed out.
Green light? I don't have a green light...Unless you mean the status light? That has nothing to do with it. The white wifi icon in the picture up near the top right of my screen shot is what I meant.
Greyed out icon screen shot:
http://i232.photobucket.com/albums/e...re14_22_49.jpg
It's there but it's not connected.