That's up to you. You are paying both halves of social security and medicare, and not getting statutory benefits like unemployment insurance. On the other hand, they might want to pay you a lower salary if they classified you as an employee.
By the way, they can't legally provide you with tax-free health insurance if you are not an employee—anything they provide of value, including paid or partially paid insurance, must be included on the 1099-MISC as part of your taxable compensation.
It sounds like you know that based on the rules, you should probably be an employee.https://www.irs.gov/businesses/small-businesses-self-employed/independent-contractor-self-employed-o...
