I'm guessing you're using OS X 10.11, you didn't specify which OS you're using - that would be helpful as well.
I've experienced this as well, the dialog you're seeing is likely feedback from your Mac that it's recovered from a kernel panic. As to the "looping", I experienced that just last week as well, in a Standard User Account - I never "work" in an Admin account, only update apps and administer my Macs, but that's besides the point here.
I "fixed" the problem by choosing the "Cancel" option, allowing me to get into the affected Account, then resetting my login Keychain in Keychain Access, restarting the affected Mac, logging back into the affected Account, then reentering my accounts and passwords - no problems since.
I can't offer any more advice, I've seen apps that access my login Keychain crash my Mac - until I reset my login Keychain. I upgraded from Yosemite, and I'm assuming/surmising/guessing that 10.11's new built-in protections may be preventing apps from accessing the Keychain items.