  We prove that for each discriminant $D \equiv 0,1 \mod 4, D \not\in\{4,9\}$,
the corresponding Prym eigenform locus discovered by McMullen in the stratum
$\mathcal{H}(6)$ is connected. Thus, the projection of any of those loci in the
moduli space is a single Teichm\"uller curve. Along the way, we obtain a
classification of primitive square-tiled surfaces in the locus
$\mathrm{Prym}(6)$ of Prym forms in $\mathcal{H}(6)$.
