You are paying this individual to do specific work at a set rate per hour because he has expertise that you need on a temporary basis. You are not his employer. He is asking for a non-industry-standard perk that is not in line with users of contract labor.
It is possible that he is new to being a contractor and does not really understand what the role is and what it is not. I hope you have a solid contract in place with him and are careful to abide by the IRS rules regarding this individual.
Otherwise, you might discover that he expects other non-standard considerations when his contract ends. (For example, it is not uncommon for a worker whose contract period has ended to apply for nemployment insurance benefits with a state agency. Then, when officials do not find any earnings reported for the worker, an investigation ensues. For misclassifications, the payouts most often involve benefits and lost wages including unpaid overtime.)
Contractors should all have specific self-determining kinds of work to do, normally in a finite time period, with defined deliverables, using their own place of work, and so forth. If you look at Section Two of the IRS Publication 15-A, you will see exactly what the IRS considers when determining whether a worker is an employee or an independent contractor (i.e., 1099 worker).
In addition, the IRS will help an organization free of charge to determine the appropriate classification for a worker or group of workers in the organization as regards federal law. To obtain that service, you simply complete and submit IRS Form SS-8 to the IRS. However, your state may have more or different requirements.