Interesting question in the first day of macro.  Maybe a Trump-effect?  First off, immigrants who are legally permitted to work are eligible for unemployment insurance.  Second, the Bureau of Labor Statistics does report unemployment numbers for the total population regardless of their immigration status.  Thus, a release of May 2016 notes that foreign-born unemployment is lower than for native-born (the survey cannot tell whether documented or undocumented):

The unemployment rate for foreign-born persons in the United States was 4.9 percent in 2015, down from 5.6 percent in 2014, the U.S. Bureau of Labor  Statistics reported today. The jobless rate of native-born persons fell to 5.4 percent from 6.3 percent in the prior year.  Data on nativity are collected as part of the Current Population Survey (CPS), a monthly sample survey of approximately 60,000 households. The foreign born are persons who reside in the United States but who were born outside the country or one of its outlying areas to parents who were not U.S. citizens. The foreign born include legally-admitted immigrants, refugees, temporary residents such as students and temporary workers, and undocumented immigrants. The survey data, however, do not separately identify the numbers of persons in these categories. Highlights from the 2015 data:

-In 2015, there were 26.3 million foreign-born persons in the U.S. labor force,  comprising 16.7 percent of the total.

-Hispanics accounted for 48.8 percent of the foreign-born labor force in 2015 and Asians accounted for 24.1 percent.

-The median usual weekly earnings of foreign-born full-time wage and salary  workers were $681 in 2015, compared with $837 for their native-born  counterparts.