Short answer, most of the time, such reimbursement would be treated as wages and subject to regular withholding of social security, medicare, state and local income taxes, as well as any rules on unemployment tax, worker comp, or anything else that is calculated based on the wage base. Then, you have no control over what the employee does with the money, you can't force them to buy insurance. However, there are some newer options.
Longer answer is that fringe benefits are discussed in publication 15-B.
https://www.irs.gov/forms-pubs/about-publication-15-b
The Affordable Care Act really turn things upside down. If you wanted to provide conventional employer sponsored health insurance (with pre-tax premium deductions for the employee share of premiums) you must offer coverage to all similarly situated employees (like all full time employees) -- you can't pick and choose your most important employees or friends or anything else. There is a non-discrimination rule.
There is something fairly new called a Qualified small employer health reimbursement arrangement (QSEHRAs). This allows the employer to provide tax-free reimbursement for qualified health expenses (proven with receipts) including insurance premiums, with a maximum reimbursement of $6150 per year for single coverage and $12,300 for family coverage. The QSEHRA can only be funded by the employer (no employee contributions), and any unspent funds can either be returned to the employer's general account at the end of the year, or allowed to accumulate for future years, but can never be paid out directly to the employee except as reimbursement for proven expenses. Also, QSEHRAs are limited to small employers (less than 50 employees) and must be offered to all similarly situated employees.
If an employee is covered by a QSEHRA, reimbursements are tax-free, but the employee is not eligible to contribute to an HSA, and can't take a tax deduction for expenses paid with tax-free reimbursements. Also, the $6000/$12000 limit may not fully cover the cost of health insurance, depending on the type of insurance purchased.
If you don't want to provide conventional employee-sponsored health coverage, and don't want to offer a QSEHRA or are otherwise not eligible, your other option is to "gross up" the salary of the employee affected. Basically, give them a raise, and hope they use it for medical insurance. You are not required to offer this to all employees, since it's a raise, not a benefit. And you can't make the raise conditional on how they plan to spend the money. Since the money is taxable to them, they would be eligible for ACA credits and to make their own HSA contributions (depending on what insurance they buy), and they can deduct their premiums and other after-tax medical costs as schedule A itemized deductions.
