The bond will appear on the June primary election ballot.
"After veterans serve our country, it's our duty to serve them," Brown said Tuesday, adding that the housing program "will make life better for veterans for years to come."
Cameron, a Vietnam War veteran who is executive director of the nonprofit Veterans Resource Centers of America, said it is "now time for the electorate to do their job — vote for it."
California voters have approved 23 veterans' bond measures since 1943 and AB 639 had support from labor and construction groups.
Cameron said $600 million would finance construction of about 3,000 rental units, and nonprofit organizations would use it to leverage additional funding.