As an independent contractor receiving IRS Form 1099-MISC, you are considered "Self-Employed" and as such do pay Self Employment Taxes which is the FICA/Medicare taxes to which you call "social security." A S.E. person pays 15.3% on its net earnings in addition to income tax. Schedule SE computes the S.E. Tax and gets attached to your Federal income tax return (Form 1040).
My answer is not intended to be giving legal advice and this topic can be a complex area where the advice of a licensed attorney in your State should be obtained.
Your employer is charged with classifying you correctly. If you know for sure that you are classified incorrectly you should tell your employer so they have a chance to correct the problem. That is your first step. I would not advise paying for your state and federal benefits via some alternative route.
You have legal obligations to pay certain taxes and other withholdings whether you are an employee or self employed. You have to do so even if it "gets the employer in trouble."
Either demand that you be properly characterized, pay as a self employed person, or violate the law. Your choice. Note: failure to pay into unemployment, SDI and Social Security will come back to bite you if you ever need those things.
Good luck to you.
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