If you are homeless your options for a place to stay are legally two-fold: a prison cell or a homeless shelter. Both options involve subjecting oneself to the capricious power and authority of another person. You are forced to live at the mercy of others. If the shelter doesn't accept you for whatever reason, even if it's not for any wrong doing on your own part, you are legally left with one option for vagrancy, loitering, trespassing (on Public property) or any number of trumped up charges. A third option is in order to secure equal freedom to the homeless. Allowing them to build shelter on public land would do this much and diminish their burden on taxpayers.
It is clear from repeated experience that municipal and state governments are not able to provide for all homeless people. For example, homelessness peaked during the 2008 recession and so did local and state government debt the next year. Thus, municipals and states are least able to shelter and provide for the homeless when homelessness is at its worst (due in large part to a drop in property values and consumer spending). Of course there are federal homeless assistance programs under HUD, such as those contained in the McKinney Vento act, but like all federal programs they propose a one size fits all solution to a complex problem and being appropriated as discretionary spending, are subject to cuts. There is no guarantee of shelter, even in a welfare state.
The National Alliance to End Homelessness estimates that 31% of homeless people are unsheltered, and the population of unsheltered homeless is growing in 32 states. About 37 percent of homeless people are families, but the vast majority of them live in emergency shelters or transitional housing. The majority of the unsheltered population, and homeless people in general, are single individuals with nothing to lose but their dignity (through panhandling) and are responsible for no one except themselves. If they are not allowed to live on public land, such as in the woods, their only other option would be living on the streets, presenting a nuisance to businesses and pedestrians. If they are not allowed to build make shift shelters on public land they are subject to the elements; they are prone to die of hypothermia, pneumonia or suffer heat stroke. They would also be left without a place to store enough belongings to survive. Shelter, much like food and water, is a necessity. Unless you live in a very mild climate, going without shelter will be fatal. If holding a person captive without food and water is murder it would also be immoral to prevent someone from finding or building shelter. Prohibiting someone from preserving their own life is clearly a violation of moral law; therefore, municipals violate moral law whenever they prohibit the homeless from building shelters on public land. Of course there are reasonable exceptions such as public parks that are readily accessible, but secluded woodlands should be fair game for homesteading.
Since all people move and have their being on land and cannot preserve their own lives without it, the right to use the earth logically follows from the law of equal liberty. A just society would afford everyone an equitable share of its natural opportunities, compensating those who are denied these privileges and ensuring that the value generated from them are put towards the public good. Furthermore, that this privilege could not exist without the support of a society seems obvious enough. Thus, they should be contingent upon both their utility to society as a whole and their direct beneficiaries. The homeless, being denied an equitable share of society's natural opportunities, should either be compensated for the value of the natural opportunities they have been excluded from or permitted to use enough of them to gratify their desires. The former resolution would take the form of a supplementary income called a citizen's dividend. The latter would permit homesteading of public land.