A question about jobs

I've read about the current state of the job market.  The outlook isn't really good.  We have pretty bad unemployment by anybody's measure.  The time it takes to find a job is as long as it has been in years (45 weeks average). It is well know that the unemployment number reported is an under-counting of the people that are idle. One thing I don't know for sure but assume is the case the unemployment number is generated by jobless claims and to claim unemployment benefits (or claims) one has to be a us citizen.  This leads to the roots of my question.  How does the illegal immigrant (II) issue play on the unemployment situation?  Let my clarify: as I say unemployment situation I mean the number of citizens that need work or idle (whether counted officially or not).  Now another point of clarification: let's discuss the population of unemployed (or idle) people.  I don't know the exact numbers but a quick search shows the population of unemployed is14 mil. and of that 7 mil. have an education past HS or some training (group Y) and 7 mil. don't (group Z).  The question of II pertains to the group Z population.  This assumes the most II fit in the group Z.  These earn a minimum of $15K/yr (based on the min. wage) and could earn say an average of $32K.  The estimates of II are between 11 mil and 20 mil. and of that a minimum 8 mil.are workers.  Next if the II are idle they wouldn't receive any income so I assume most are employed in some way and if not they wouldn't be motivated to come here (cost of living is higher than in their home countries).  But some could be working part time or temporarily idle so I will say that 7 mil work (90% of the estimate of workers) thus 7 mil x $15K is $105 bil.  So let's list the major assumptions:
Unemployed means idle (not the reported number which is a structured under counting)
The unemployed 14 mil is divided into 2 equal groups Y (educated/skilled) and Z (uneducated/unskilled)
there are 8 mil II workers in the country
II earn a minimum of $105 bil

Now the arguments for and against this situation come into play. The first one against II. Given the assumptions: if no II were here than that 7 mil more citizens could be employed reducing the unemployment numbers by half and helping the economy.  Wiping out the group Z unemployed.  Is it a great coincidence that the numbers of II workers equal the number of unemployed citizens (HS education or less)?  Remember a portion of $105 bil is remitted to their home countries and out of the US economy.   One argument for II working is: citizens won't work in the jobs   that the II do (agriculture, construction, domestic, etc).  These industries are also ones where the group Z would qualify manual labor, low skill needed thus leading to an assumption. This just scratches the surface of the problem but the question remains: How does the illegal immigrant issue play on the unemployment situation?