Congratulations on at least getting to speak to someone, many people do not even make it that far
"Eligibility passed" means the Visa Officer reviewed your eligibility and considered it to be satisfactory. It all comes down now to the Criminality and Security assessment. My guess is criminality probably is passed too (as it's often started early on) but Security is likely to be in progress still.
Next time you call, don't ask about background check but specifically ask about the Criminality and Security assessments, as these things make up the Background check.