Does an employer have the right to perform random **** testing if it was not mentioned it could happen at the time of hire?

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 has increase/decrease salary/hours, promote/demote, and generally 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.