You do not have to sign a agreement after employment that requires you to waive your right ot a jury trial. It is a different story if you are presented with the agreement prior to working there. You should consult with a labor or employee rights attorney. The Orange County Bar Association has a referral service. As part of that service the attorneys must give you a free consultation. I am sure you can get these questions answered by a specialist in one phone call.
California is a "at will" state. You are already "at-will" unless you have a written agreement which states that you are employed for a certain periord of time. If you have no such written agreement you are at will.
Disparage means you will not bad mouth the employer, say things to others that will hurt the employer's reputation or cause harm to the employer.
In a word, yes. Unless you have a written employment agreement that grants you the right to a trial by jury, or you're in a union that has a contract with your employer that has such a clause, your employer can require you to sign a waiver. Without a written employment agreement, you're considered an "at-will" employee, which means that you can be fired for any reason, or for no reason at all. You can be fired because your boss got out of bed on the wrong side that morning, or his kid got a bad record, or he decided he doesn't want any employees with red hair.
The caveat to this is he may not fire you if you don't sign the waiver if he only fires people who are members of what the law calls a "suspect" class - race, gender, age, disability, etc.
Such firings would violate the civil rights laws.
While it sounds wrong, your employer has the right to do this.
As for "disparagement," that means to speak badly of your employer, now or in the future. That's tricky, for it risks running afoul of your First Amendment right to free speech. You'd do well not to speak about your employer at all (except with your family) or you might be fired, and sued for slander and defamation. While this is what the courts' mean by a "chilling effect" on free speech, such a suit against you would cost you a fortune to defend, and it sounds like your employer already has lawyers lined up. If you lost, you could lose your house and all your other assets.
Contact your union, if you have one. They have lawyers, too. Explain what's going on, and see what they advise you. In the meantime, I suggest that you not discuss your employer with your co-workers (you never know who might be reporting back to management) and just do your job. Sign the waiver or you'll be fired. The waiver doesn't say that you're waiving a trial; you're just waiving a jury. It offends me, as a lawyer, but there's really nothing you can do.
DISCLAIMER - THIS IS NOT LEGAL ADVICE, AND NO ATTORNEY-CLIENT RELATIONSHIP IS CREATED. THIS IS FOR GENERAL INFORMATION PURPOSES ONLY.