Well, yes and no. There are several factors. First of all, government can both help (by providing money/actual housing) and impede (e.g. by requiring ID checks when renting out). Help may include giving money (or pseudo-money, such as vouchers) to participate in the private market.
Second, many illegal immigrant are poor, or comparatively poor (e.g. housing in SF Bay Area is crazily expensive as you know, but in other parts of California is not super-cheap either usually) - and thus the government has a choice - either provide at least some assistance, or deal with tons of homeless people and all the problems that arise from that. California has pretty mild climate, which can be good because people won't freeze to death, etc. in most places most of the time, but that also means they can actually live on the street in most places most of the time, and the rest of the population usually gets uncomfortable when this happens. So they ask the government to solve this somehow.