Can the insurance company deny your claim if another person crashes your car and they think they live with you and are not on the policy?

Absolutely. If they are a household resident or if they are a regular driver they must be listed on the application for insurance. You have a contractual obligation to notify the insurance company if residence and drivers. If you don't then you are in violation of the contract and have committed material misrepresentation. This gives the company the right and obligation to deny the claim. An insurance application and policy are legal binding contracts. If you fail to comply with the terms of the contract in order to cheat the company out of premiums why should they comply with their terms and pay the claim. Insurance companies are audited by the State Department of Insurance and get in trouble for paying claims they shouldn't just as they will for not paying claims they should.