Can an employer make you take a snow day when work is closed as a personal day?

Yes, it can.  An employer has a right to do this if you are an "at will" employee.  That is if you do not have an employment contract or union agreement that prohibits this action.  Basically, in an at will employment situation, an employer can hire or fire someone for a bad reason, or any reason, or even no reason at all.  Additionally, it can increase or decrease an employee's salary or hours, , promote or demote an employee, and generally impose requirements as it sees fit.  In turn, an employee can choose to work for an employer or leave.  This holds true unless there as been some type of workplace discrimination involved.   

A personal day is not something that an employee is automatically entitled to.  As a general rule, there are no laws governing such time.  They do not have the right to use any unused time whenever they please.  The reason is that it is up to an employer's discretion as to whether or not to provide such a benefit.  Consequently, an employer may deny a request for personal time or it may approve it, as well as mandate when such time can or must be taken.