I filed my forms 843 for refund for 2006 and 2007 this year in April. I have since received three letters from the IRS asking me to give them 45 more days to work on my claim. When I called to find out what was going on, I was informed that nothing was being done with my claim and that it had not been worked on. I have since given power of attorney to a CPA to look into my claim. I am on a J-1 visa working as a teacher since August 2006. What are my chances of getting my FICA back?

J1 teacher, professor and researcher are exempt from residency for two years and are not subject to FICA taxes. So you do not pay FICA taxes for 2006 and 2007, and you must file non-resident tax return.
Also your income may be exempt from federal income tax for period as per tax treaty.
This year 1RS have been over busy due to stimulus rebate checks. So you must wait.
Read about filing Form 843: Your U.S. Tax Return: Social Security and Medicare (FICA) Taxes for Non-resident Exempt Individual

All things being equal, you WILL get the refund back .

You may want to give the IRS International Tax Hotline a call at 215-516-2000 to check on the status. They are less likely to give you any run around.