Why is the error square equivalent to variance of error?

I'm confused as to why is V($$\hat β$$) = E[$$\ εε^t$$] What's an intuitive explanation for this?

Additionally, is this the reason why when we run Breusch Pagan test, we regress $$\ ε^2$$ on independent variables because we are implicitly running variance on independent variables?

Thank you !

Therefore, you are left with the squared term. $$Var(\epsilon) = E[\epsilon^2] - E[\epsilon]^2 = E[\epsilon^2]$$ because $$E[\epsilon]$$ = 0.