It is a myth (at least in 2019)

> **A significant portion of homeless people do have jobs**—they just cannot afford to pay rent. Some receive disability income due to physical or mental problems but still cannot afford rent. **For those wanting to work—a common refrain among those interviewed by my research team**—the complications of applying for a job with no address, no clean clothes, no place to shower, and the stigma of being homeless (or having a criminal record), make such individuals far less competitive in the low-wage job market.

(emphasis mine)

https://www.nyu.edu/about/news-publications/news/2019/september/HomelessQandA.html

Wikipedia hints that working dost not prevent homelessness and some of the major causes of homelessness impact workers:

 - Lack of sufficient urban housing projects to provide safe, secure,
   and affordable housing to the financially underprivileged.
   Additionally for low wage workers rents can be unaffordable in areas
   where their workplace is located. 
 - [...] 
 - Redevelopment and
   gentrification activities instituted by cities across the country
   through which low-income neighborhoods are declared blighted and
   demolished to make way for projects that generate higher property
   taxes and other revenue, creating a shortage of housing affordable to
   low-income working families, the elderly poor, and the disabled.
 - [...] 

https://en.wikipedia.org/wiki/Homelessness_in_the_United_States#2001%E2%80%932020