Prime numbers and other numbers all have square roots. All real numbers like 1, 2, 3.444444, 5/6 , √5, 9 , 0197 -- have all square roots. The number has to be a positive number or zero to have a square root.
For some numbers the square root is an integer or a rational number.
For some numbers, square root is irrational. You cannot write it as p/q like 4/5...
Prime numbers like 2, 7 , 11, 127 all have square roots, but they are not written as 2.3389484.. as their decimal digits are endless. So we write them as
√2, √3 , √7, √127 like that.