We can solve the hunger problem and the homeless problem in one swoop!
I don't know where this rumor about homeless being "disappeared" started, but it sounds like a typical Urban Legend.
The OP mentioned Boston. I'm there quite a lot. There are plenty of homeless on the streets there and in the other urban communities here. But in the winter, the police cruise the streets with a van or "wagon" and will ask any homeless person they see whether he or she would like to go to a shelter for a meal, hot shower and bed. The next day they disperse again. When the weather warms up, fewer will want to go to a shelter at night, so you see more of them on the streets after dark.
Those with mental illnesses often refuse to go to a shelter, as do others who don't want to go to a shelter because they prefer to be by themselves, and because shelters are usually crowded and even overcrowded, with not enough staff to help and to keep an eye on things. You can end up sleeping on a mat on the floor, and there is a real risk of being robbed or bullied, even beaten, at shelters. Unless they are "disturbing the peace" they will be left free. I have seen the police cavassing homeless street people and squatters and then leaving them be when it was obvious they didn't want to go to the shelter.
Also, it is not unusual for an entire city to attempt to divest itself of homeless people and other "undesireables" when a big event such as a convention, or the Olympics, is going to be held there. Remember the Atlanta, GA Olympics? Atlanta gave one-way bus tickets to all of the people it didn't want to be there, and "invited" them to leave town. This does happen in a lot of places. But those people have to go somewhere, and they do not really disappear. They just show up in some other city or town before being sent on again.
What a suck way to have to live.