All three conditions can be cast as conditional statements:
- Reflexive:$\;$If $a\in X$, then $aRa$.$\\[4pt]$
- Symmetric:$\;$If $a,b\in X$ and $aRb$, then $bRa$.$\\[4pt]$
- Transitive:$\;$If $a,b,c\in X$ and $aRb$ and $bRc$, then $aRc$.
For any set $X$, if $R$ is the empty relation on $X$, the hypothesis is false for the symmetric and transitive conditionals, hence those statements are true.
For the reflexive conditional, if $X\ne\large\varnothing$ and $R$ is the empty relation on $X$, then by choosing $a\in X$, the hypothesis is true but the conclusion is false, hence the statement is false.
Thus if $X\ne\large\varnothing$ and $R$ is the empty relation on $X$, $R$ is symmetric and transitive but not reflexive.
However if $X=\large\varnothing$ and $R$ is the empty relation on $X$, the hypothesis for each of the three conditionals is false, hence those statements are true.
Thus the empty relation on the empty set qualifies as an equivalence relation.