Your employer likely has the right to terminate your employment at anytime though a layoff. That's because employment is generally presumed to be "at will" in the U.S. It's also because employers have the right to protect themselves financially through layoffs.
However, you have at least some employee rights regarding a layoff. For example, your employee rights might entitle you to collect state unemployment benefits and purchase COBRA extended health insurance benefits at group rates after you're laid off.
Your employee rights also protect you from your employer illegally firing you under the cover of a layoff. Read Wrongful Termination for information about the types of employment discharges that might be illegal under the cover of a layoff.
Additionally, your employee rights might require your employer to give you advanced layoff notice, so that you may at least prepare financially to lose your job while searching for another during your layoff notice period.
Under certain circumstances, the Worker Adjustment and Retraining Notification Act (WARN Act), a Federal "layoff law" so to speak, requires employers to give affected employees up to 60 days of advanced notice for mass layoffs or plant closings.
Under the WARN Act (or an equivalent state law generally referred to as a "mini-WARN Act"), your employer must continue to pay you and grant you the benefits to which you're entitled through your advanced layoff notice period, whether or not your employer requires you to work through it.
However, if you start a new job during your layoff notice period, under the WARN Act it's the same as resigning from your current job. If your employer discovers that you've started a new job, then your employer likely will not owe you for the remainder of your layoff notice period.
U.S. district courts enforce the WARN Act. If your employer is required to, but fails to honor WARN layoff notice requirements and you wish to seek damages, then WARN permits you (or your union) to file a lawsuit against your employer in a district court. Consult an attorney about filing a WARN Act lawsuit. WARN authorizes the court to grant you reasonable attorney's fees as part of your settlement.
Final pay laws vary by state; but, generally, employee rights under such laws require employers to issue final paychecks immediately or soon after employment ends, such as by the next regularly-scheduled payday after a layoff effective date.
Although your employer might offer severance pay anyway, your employee rights do not generally entitle you to receive it for a layoff, unless you're working under a collective bargaining agreement, or an explicit or implied contract that indicates otherwise. The pay you'd receive through the end of your layoff notice period under the WARN Act is your regular pay in compliance with the Act's advanced notice requirements, not severance pay by definition.
Contact the relevant state labor department or consult an attorney for more information about employee rights regarding a layoff.