Crowns in pseudo-random graphs and Hamilton cycles in their squares

A crown with $k$ spikes is an edge-disjoint union of a cycle $C$ and a matching $M$ of size $k$ such that each edge of $M$ has exactly one vertex in common with $C$. We prove that if $G$ is an $(n,d,\lambda)$-graph with $\lambda/d\le 0.001$ and $d$ is large enough, then $G$ contains a crown on $n$ vertices with $\lfloor n/2\rfloor$ spikes. As a consequence, such $G$ contains a Hamilton cycle in its square $G^2$.