Not exactly. From the US Bureau of Labor Statistics:

Is the count of unemployed persons limited to just those people receiving unemployment insurance benefits?

   No; the estimate of unemployment is based on a monthly sample survey of households. All persons who are without jobs and are actively seeking and available to work are included among the unemployed. (People on temporary layoff are included even if they do not actively seek work.) There is no requirement or question relating to unemployment insurance benefits in the monthly survey.

Does the official unemployment rate exclude people who have stopped looking for work?

   Yes; however, there are separate estimates of persons outside the labor force who want a job, including those who have stopped looking because they believe no jobs are available (discouraged workers). In addition, alternative measures of labor underutilization (discouraged workers and other groups not officially counted as unemployed) are published each month in the Employment Situation news release.

Are undocumented immigrants counted in the surveys?

   Neither the establishment nor household survey is designed to identify the legal status of workers. Thus, while it is likely that both surveys include at least some undocumented immigrants, it is not possible to determine how many are counted in either survey. The household survey does include questions about whether respondents were born outside the United States. Data from these questions show that foreign-born workers accounted for about 15 percent of the labor force in 2006 and about 47 percent of the net increase in the labor force from 2000 to 2006.

My own opinion would be that undocumented ("illegal") workers are significantly undercounted because of their (justifiable) fear of the consequences of disclosing their presence.