If the false update alert actually refers to the name of the game you're trying to play, then I doubt very much that it's caused by malware or a compromised router. It's just a defective website.
It's not for me to tell you how you should spend your leisure time, but honestly, if you want to play games, you can do a lot better than what's on the "webkinz" site. There are thousands of games in the Mac App Store, for example, and I would imagine that even the worst of them is better than the content of that site.
More importantly, Flash is not entirely trustworthy. Every couple of months, it seems, a new security exploit in it is discovered, and Apple blocks all old versions of Flash, forcing everyone to update. This has happened many times, and it shows no sign of coming to an end. That means that even when Flash is up to date, there's a chance that it may have an as yet unknown vulnerability that someone could take advantage of. Playing Flash games exposes you to the highest risk. I strongly suggest that you avoid using Flash content unless it's really necessary.