I have plan that has a 3 month wait for eligibility. The client has several employees that are paid and receive a 1099 form rather than a W-2. The plan includes all employees except: Union Employees, Nonresident Aliens, and Leased Employees.
Do I need to include the employees who are paid and receive a 1099 form? Are these employees considered Leased Employees?

You should not consider them as being anything without further investigation.

They could be statutory employees, they could be misclassified employees, they might really be Independent Contractors or most likely they are employees of the leasing company.

I would question the giving of 1099s to leased employees. Most leased employees are employees of the Leasing company. Was payment made to these employees or to the leasing company?

If made to the employees, with whom was the leasing arrangement? If made to the employees and there is a leasing arrangement with the employee, they most likely are not ICS, but it is the case they very likely are misclassified. It is questionable that a company could directly enter into a lease arrangement with the individuals themselves.

I would not assume anything about these employees until you find out exactly what they do and whether or not they meet the common law test for an employee.
The IRS frowns on issuing 1099's to common law employees. This is not a choice that an employer can make for convenience or to avoid the payment of employment taxes. Leasing is another issue entirely--again you need to consider all of the facts. Are the leasing parties related? Is this a bona fide lease?