If I do a bank deposit for a company which they pay me mileage to do,do they have the right to ask me for my proof of insurance.

Yes, I am not aware of any jurisdiction in which automobile insurance information is protected from disclosure.  You can refuse to provide it, but then they can fire you.  The company obviously has an interest in confirming that you have auto insurance, since if you get into an accident while performing services for the company, it may be liable.