Your boyfriend committed the cardinal sin of embarrassing the HR/Controller and that is almost certainly the reason she was determined to find a way to get rid of him. Unfortunately, assuming he is, what we call, an "at-will" employee, he can be terminated for almost any reason, as long as it is not an unlawful one. And your post does not identify an unlawful reason.
Now, if this HR/Controller were to publicly accuse your boyfriend of theft, he might have a claim for defamation. When I say publicly, I mean to a third party, such as a prospective employer. It does not include the EDD. However, if the employer does say they fired your boyfriend for even the perception of theft, the burden of proof is on the employer to prove that he actually did steal something, if they were to succeed in denying his benefits.
They say you get what you pay for, and this response is free, so take it for what it is worth. This is my opinion based on very limited information. My opinion should not be taken as legal advice. For true advice, we would require a confidential consultation where I would ask you questions and get your complete story. This is a public forum, so remember, nothing here is confidential. Nor am I your attorney. I do not know who you are and you have not hired me to provide any legal service. To do so would require us to meet and sign written retainer agreement. My responses are intended for general information only.
Unemployment is likely. Any other claims will not work, as California is an at-will state, and the emloyer can terminate even if they are undeniably wrong on the basis for termination.
Please check the law of your state