I was on the BT site checking my latest bill, I went to the section for BT Fon and I saw that the WiFi hotspots rely on using some of my Broadband. I'm not worried about security or how much slower my connection is, but if this accounts for the flashing light on the Modem, (White box) when I'm not actually online? Any ideas please.

Depending on what icon you are referring to, there should be flickering on any router as it sends/receives data. What icon is it and what model of modem is it?

The white box is only the modem. It doesn't have WiFi on it, that s on your router.