Wow, I haven't had any problems logging into facebook and it would be scary to get a message like that.
What browser are you using, Firefox, Google, Safari? I have all three of those on my laptop and I mostly use Firefox and Google but I still use Safari at times too even though I was told it was outdated. I think some folks still use IE (Internet Explorer) which is apparently waaaaay out of date. Maybe you could try logging onto FB by downloading and using another browser?
I googled and found this from the Facebook Help Center:
https://www.facebook.com/help/...
I am really technically challenged so hopefully someone with computer knowledge will see this and be able to offer advice.