The definition of a prime number is one in which there are no factors other than 1 and itself. For example, 5 has no other factors other than 1 and 5 (itself).
The number 1 is not considered prime nor composite.So, even though it is a square (1²), it is not a prime number.
All square numbers have a square root that is neither 1 nor itself: 16 is the square of 4, 49 the square of 7, and so on.
Due to the fact that square numbers have factors other than 1 and itself, square numbers are not prime.
Hope this helps! :)