For any smooth Riemannian metric on an $(n+1)$-dimensional compact manifold with boundary $(M,partial M)$ where $3leq (n+1)leq 7$, we establish general upper bounds for the Morse index of free boundary minimal hypersurfaces produced by min-max theory in the Almgren-Pitts setting. We apply our Morse index estimates to prove that for almost every (in the $C^infty$ Baire sense) Riemannan metric, the union of all compact, properly embedded free boundary minimal hypersurfaces is dense in $M$. If $partial M$ is further assumed to have a strictly mean convex point, we show the existence of infinitely many compact, properly embedded free boundary minimal hypersurfaces whose boundaries are non-empty. Our results prove a conjecture of Yau for generic metrics in the free boundary setting.
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