By 'SSI' do you mean 'social security income' or 'supplemental security income'.

If the former, than French citizenship will not cost you your social security benefits.

However, if the former:

WHO  IS  ELIGIBLE  FOR  SSI ?  

Anyone who is:

• aged (age 65 or older);
  
• blind; or
  
• disabled.

And, who:
        ...

• is a resident of one of the 50 States, the District of Columbia, or the Northern Mariana Islands; and
  
• is not absent from the country for a full calendar month or more than 30 consecutive days;

Understanding Supplemental Security Income (SSI)-- SSI Eligibility