Is the language of words that contain a square regular or context-free?

A word is square-free if it contains no non-empty sub word of the form $y^2$. Your language consists of all words over ${a,b}$ which are not square-free. It is not difficult to enumerate (using exhaustive search) all square-free words over ${a,b}$:
$$
epsilon, a, b, ab, ba, aba, bab.
$$

Since this list is finite, the language of square-free words over ${a,b}$ is trivially regular, and so is its complement.

In contrast, the language of square-free words over ${a,b,c}$ is not regular. This follows from the existence of an infinite square-free word $w$, which can be obtained from the Thue–Morse sequence.

Indeed, I claim that the set of prefixes of $w$ is pairwise inequivalent modulo your language. To see this, let $x,y$ be two such prefixes, say $x$ is a prefix of $y$. Then $y = xz$, and so $xz$ is square-free while $yz = xz^2$ isn’t.

Alternatively, we can apply the pumping lemma to the language of all square-free words over ${a,b,c}$. If the pumping length is $n$, take the prefix of $w$ of length $n$, and pump it up so that it contains a square. Similarly, since the Thue–Morse sequence is an infinite binary cube-free sequence, the language of cube-free words over ${a,b}$ is not regular.

The latter argument shows that the language of square-free words over ${a,b,c}$ and the language of cube-free words over ${a,b}$ are not context-free. This leaves the following questions open:

  1. Is the language of non-square-free words over ${a,b,c}$ context-free?
  2. Is the language of non-cube-free words over ${a,b}$ context-free?