Which (possibly implicit) assumptions and conclusions (that later turned out false) made it hard for Cantor to believe that there is a bijection between the unit interval $[0,1] \subset \mathbb{R}$ and the unit square $[0,1]^2$, i.e. $|[0,1]| = |[0,1]^2|$?
It was all clear to him that there is a bijection between $\mathbb{Z}$ and $\mathbb{Q} \equiv \mathbb{Z}^2$ and therefore between $\mathbb{Q}$ and $\mathbb{Q}^2$, i.e. $|\mathbb{X}| = |\mathbb{X}^2|$ for some countable infinite sets (even when not enumerable in natural order). He also knew that $|\mathbb{Q}^\sqrt{}| = |\mathbb{Q}^{\sqrt{}} \times \mathbb{Q}^{\sqrt{}} |$ for the "Euclidean" numbers (mainly because $\mathbb{Q}^\sqrt{}$ is countable) and probably that $|\mathbb{X}| = |\mathbb{X}^2|$ for all countable sets $\mathbb{X}$.
But his assumption must have been that the latter doesn't necessarily hold for uncountable sets. If he had not thought so, he would not have been surprised to find that there is a bijection between $[0,1]$ and $[0,1]^2$, but he seemed to be so:
But for what specific reasons did he believe that $|\mathbb{X}| = |\mathbb{X}^2|$ does not necessarily hold for uncountable sets $\mathbb{X}$?
It seems to me that the answer to your question is already contained in the article you linked to. Cantor says that he originally didn't expect this because it was widely believed that it takes $n$ coordinates to specify a point in an $n$-dimensional manifold. (He also says that others held this to be self-evident whereas he believed it required a proof.) By the way, the article argues that his remark "I see it, but I don't believe it." doesn't actually, as you seem to imply, refer to the result but to the proof.