Does the mechanical engineering program require prerequisites before actually entering into the program?

No prerequisites that I am aware of, but high school calculus and physics will be very helpful in starting out. Also, many schools won't require you to declare a major until a few semesters in, as many of the first several core engineering courses are the same for everyone. (Calculus, physics, etc)

This is specific to the school you attend however a lot of engineering programs require prerequisites of Math and the sciences. I had to complete calculus and physics before I could take the core engineering classes at my school.