easy enough. Given five consecutive numbers , , , and , for their product to be a square they need to combine all their prime factors with even powers. For instance, taking only two numbers for simplicity, for their product to be a square we need for instance or , and is not a square because every prime factor doesn't have an even power.
Now assume some prime number greater than three divides . It obviously doesn't divide , , , . Therefore, this prime factor is only present once in the product , hence has an odd exponent (or power) ( ) and thus is not a square.
Now just validate the statement with the two prime numbers smaller or equal to three, namely 2 and 3 (just take and , since any multiple is suitable), to conclude the proof. Oh, and also validate for and ! And for negative numbers, you only have to check up to , because the product of five negative numbers is a negative number which cannot be a square !
Does it make sense ?