Housing Homeless Veterans Act of 201 7 This bill amends the United States Housing Act of 1937 to require the Department of Housing and Urban Development (HUD), in making Veterans Affairs-Supported Housing (VASH) program funds available among public-housing agencies (PHAs) for rental voucher assistance for homeless veterans who have chronic mental illnesses or chronic substance-use disorders, to give priority to PHAs that: (1) serve areas having the highest numbers of homeless veterans, and (2) have fully utilized their available HUD-VASH funds for the preceding fiscal year and have demonstrated need for such incremental assistance. HUD shall: (1) recapture from a PHA certain unused HUD-VASH assistance, and (2) reallocate recaptured amounts to PHAs that have used their entire allocations of HUD-VASH assistance.