Unrelated to this issue, I got an Apple Time Capsule, and it said "Double NAT error", making me look into this error (iCam Pro still didn't work with the Time Capsule replacing the Linksys router). Turns out that my ISP had started providing dynamic ip addresses, resulting in no incoming connection. I asked my ISP for a static ip address and that seems to have solved the problem. iCam Pro is able to connect again now.
If anyone else find this thread and have the same problem, here's some info about my situation: I'm in Denmark. I have Fibia/Waoo as an ISP. I called them to get a static ip address (well, technically not static, but a "locked" ip so that I get the same ip all the time from their DHCP-server).