Thank you for the information and your questions. I'll address each of them below.
1) Can they fire him for a cause that never occurred? (I have been told in Delaware you can be fired without cause but if fired for cause then it must be a truthful cause. Your understanding of the law is partially correct. The employer can terminate for any, or no, reason and with notice or warning unless a contract states differently or they terminated him because he is in a protected class under employment discrimination laws. There is nothing in the law that says they cannot be arbitrary or incorrect about the termination. However, as touched on below, if there was no valid basis for a "cause" termination, he will likely have success with his UI claim and could sue them later for defamation of character if they make a false statement of fact to a prospective employer.
2) Can they stop him from getting unemployement and if not, what should he do? No, an employee cannot legally waive their right to file for UI benefits. That is against public policy. So, he should apply and be prepared to appeal if his former employer tries to fight his UI benefits.
3) How can he be sure that this will not effect his ability to get other jobs...will his record with this food chain claim he was fired for theft? He worked there for 3 years with great reviews and he must use this as a reference
He can't be sure, but as mentioned above if he knows or suspects they are making false statements, he can file suit for defamation.
4) Should he hire a lawyer to at least write a letter asking for his job back or at least having his record changed to fire for no cause and get appropriate OK for unemployment? FYI. His boss was fired less then one week from firing him. 3 Aug
It wouldn't hurt to try to put pressure on the employer through an attorney. It doesn't mean they will change their stance, but it can't hurt.
Please feel free to ask for clarification, I would be glad to assist you further if I can.