first, the number n=11...1=(*)x100+11; and therefore the remainder is 3 dividing by 4; hence n is not a perfect square since a perfect square odd number must have remainder 1 dividing by 4.

now if n=(p/q)^2 with p, q natural numbers and (p,q)=1. Then

nq^2=p^2 (1)

If q>1, let s be a prime factor of q. Then s is not a factor of p or p^2 because (p,q)=1. Then (1) does not hold. It derives that q=1.
Consequently, n=p^2 is a perfect square. A contradiction.