Asked to "seek other employment then give resignation." WTH?

You are in the United States, unless you are in a union or have an employment contract (not agreement, a "contract") you are an "at will employee". This law allows you to quit at any time with or without notice. This law also allows your employer to let go of you at any time, with or without notice.

Your employer does not need to figure out a "case" to fire you. And employers really don't care about filing unemployment claims ~ getting rid of a bad employee is way cheaper than keeping on. And in most states, the employer is already paying the top unemployment tax because the unemployment fund is bankrupt.

Your employer wanted to provide you with a bit of options...

She is giving you a chance to find a new job before he fires you. She doesn't need a case, this is not a court of law. She can just fire you.

100% legal, she's doing you a favor.

If you're in the US, she doesn't need a "case" to fire you, she can fire you for any reason or no reason. She's cutting you a break, letting you have an income while you're looking. Look hard and fast.

Yes it's legal. She doesn't owe you a job.

Yes. It is rather common in various fields of work.