No; since you were properly treated as an employee, the employer is responsible for collecting and paying over the FICA taxes. Your personal income taxes will be based on what you report from Form W-2.
If an employer fails to withhold and pay over to the government an employee’s FICA taxes, in either a current or a subsequent year the employer can make an adjustment when the error is discovered to the quarter in which the underpayment occurred.
If the employer discovers the error after the calendar year of the wage payment closes, the employer provides the employee and the Social Security Administration (SSA) a corrected Form W-2 (Form W-2c, Corrected Wage and Tax Statement) reflecting additional FICA earnings for the prior year and FICA tax withholding as if the employer had made it correctly (Regs. Secs. 31.6051-2(c)(a)–(c) and 31.6051-1(c)(1)).
This change to FICA wages and FICA tax withholding generally does not affect the employee’s prior year individual tax return. At this point, the employer has paid the employee’s taxes. The regulations provide a specific remedy to allow an employer to recover from the employee’s pay the FICA taxes that the employer paid on the employee’s behalf (Regs. Sec. 31.6205- 1(d)(1)).
If the employer does not recover the amount from the employee, the payment of the employee's share of FICA tax by the employer is current wage compensation, subject to FICA and income tax withholding, and reflected on the employee’s year-end Form W-2.
See this IRS webpage for more information regarding the employer's options for correcting employment taxes.
Please see this TurboTax tips article regarding employees and FICA taxes.
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