Thank you for the information and your question, and I am sorry to read about your situation. Unfortunately, you would not qualify for UI benefits. The reason is that you voluntarily quit your job and it was not for "just cause" attributable to the employer you quit working for. In other words, that employer would have had to do something extreme and adverse in your case that qualified as "just cause" to quit and you would have had to try to work things out with them before quitting. So, based on the information you provided, a UI claim is not an option.
As far as the job offer goes and the lack of clear communication by the new employer, unless you have an actual contract, versus an offer letter, that guarantees a start date and you did all you had to do under the contract, you would not have any legal option to force the payment of wages or work while you wait for the security clearance. If this is "at will" employment, then the fact is that the employer could have withdrawn their offer after you gave notice to the old employer, or even if you started working, for any or no reason and with no notice or warning. So, there are never any guarantees of employment unless there is a contract.
You might consider taking your offer letter to a local employment law attorney to see if there is enough detail in the letter to make it enforceable as to your start date, despite the qualifying language of the security clearance requirement. It is doubtful that it is, but it won't hurt to have local counsel take a look at it. If they think you have a case for detrimental reliance, they might suggest that they contact the new employer to try to get you paid from May 24th.
Please feel free to ask for clarification if needed. If none is needed, then if you could take a moment to leave a positive rating in the box above, I will receive credit for assisting you today. Thank you