Following from questions 2090325 and 2090607, is it possible to show that any AP with first term $a$ and common difference $d$, where $a,d\in\mathbb N$ must contain a square and, following from question 2090325, infinitely many squares? If not, then what conditions can be placed on $a$ or $d$ such that this is true?
As a partial answer to the last part, from question 2090325, we can specify that if $a$ is a perfect square then there are infinitely many perfect squares in the AP.
Addendum
Following from comments by @lulu below, here's the Wikipedia entry for Quadratic Residues.
There is an obstruction $\pmod d$. Specifically, the goal is equivalent to the statement "$a$ is a square $\pmod d$". Indeed, $$a+dk=x^2\implies x^2\equiv a \pmod d$$ Conversely, solving $x^2\equiv a \pmod d$ means there is some integer $k$ with $a+dk=x^2$.
Thus, for example, you can not have a square in the progression $\{3,8,13,18,\cdots\}$ because $3$ is not a square $\pmod 5$.