Are any prime numbers also square numbers?

No. That is impossible.

A Prime number is defined as a number that factors only to itself and the number one. A prime number MUST only be "divisible" by 1 and itself to get an integer number. For example, 7 only has 2 factors - 1 and 7.

Square numbers are divisible by a number other than 1 and itself. For example, 16 has the factors 1, 2, 4, 8, and 16. 4 squared = 16