Let q be a perfect power of a prime number p and E(Fq) be an elliptic curve over Fq given by the equation y2=x3+Ax+B. For a positive integer n we denote by #E(Fqn) the number of rational points on E (including infinity) over the extension Fqn. Under a mild technical condition, we show that the sequence {#E(Fqn)}n>0 contains at most 10200 perfect squares. If the mild condition is not satisfied, then #E(Fqn) is a perfect square for infinitely many n including all the multiples of 12. Our proof uses a quantitative version of the Subspace Theorem. We also find all the perfect squares for all such sequences in the range q<50 and n≤1000.