  Let $C: y^2=ax^4+bx^2+c$, be an elliptic curve defined over $\mathbb Q$. A
set of rational points $(x_i,y_i) \in C(\mathbb Q)$, $i=1,2,\cdots,$ is said to
be a sequence of consecutive squares if $x_i= (u + i)^2$, $i=1,2,\cdots$, for
some $u\in \mathbb Q$. Using ideas of Mestre, we construct infinitely many
elliptic curves $C$ with sequences of consecutive squares of length at least
$6$. It turns out that these 6 rational points are independent. We then
strengthen this result by proving that for a fixed $6$-term sequence of
consecutive squares, there are infinitely many elliptic curves $C$ with the
latter sequence forming the $x$-coordinates of six rational points in
$C(\mathbb Q)$.
