An employer can terminate for any reason that does not discriminate based on protected classes such as age and gender. If the employer told her she could quit and collect unemployment this is not correct. A voluntary termination will not allow for unemployment benefits.
If your mother is denied unemployment she can relate the facts of her resignation, and in dictate that the resignation was not voluntary, and obtained by fraud.
If your mother believes there is a other reason based on age (over 40) or other discrimination she should contact the Equal Employment Opportunity Commission and file a claim.
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