If $x_i$ is a particular outcome of the random variable $X$, and $\mu$ is the mean of the distribution of $X$, then is it true that $E[(x_i-\mu)^2] = Var(X)$?
Maybe I should ask a different question: is it true that $E[x_i]=E[X]$? $x_i$ is already known to us - that's the source of my confusion. We cannot expect anything from something we know.
It is common to write random variable as upper case letters and realizations as lowercase. Following this standard, and as you say, $x_i$ is a realization or outcome. $x_i$ is simply a number, not a random variable.
$E(x_i)=x_i$ which is only equal to $E(X)=\mu$ by coincidence.
Similarly, $E[(x_i-\mu)^2] = (x_i-\mu)^2$ this will only be equal to $Var(X)=\sigma^2$ by coincidence.