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...
check this link, it will enable you to determine your true status
you can file ss-8 to let the IRS determine your status
here is a link to a 20 point checklist. read what it says at the top
Misclassified Workers Can File Social Security Tax Form
Workers who believe they have been improperly classified as independent contractors by an employer can use Form 8919, Uncollected Social Security and Medicare Tax on Wages to figure and report the employee’s share of uncollected Social Security and Medicare taxes due on their compensation.