Can my employer require me to take the flu shot?

Yes, your employer can do this.  The reason - most employment relationships are what is known as "at will".  This means that basically the employer can hire or fire someone for any reason or no reason whatsoever, as well as impose requirements as they see fit.  In turn, you can work for an employer, or not, your choice. 

The only exceptions to the above would be if there is a stated company policy covering this, or there is a union/employment agreement that governs, or this situation has arisen due to some type of discrimination.