If you are a 1099 wage receiver then the obligation is on you if you decide to contribute to FICA taxes otherwise it is not the employer's fault since YOU choose that method of payment.
Further more if the employer is classifying you as a 1099 contractor incorrectly when you should be an employee he is subject to make everything good including Fica taxes, UI and workerscomp.
Obviously you had no issues not paying FICA taxes now did you.
And just to let you know if you are a 1099 employee you would have been required to file a W-9 instead of a W-4.
First off if one is receiving a 1099 one has given the contractor a SS# or a FID#. If you have a FID# then you should be paying taxes or that contractor will be getting a call from the IRS. Those 1099#'s are sent to both the state and feds.