Latex \in and \Z both cause Undefined Control Sequence error

\in is a math mode symbol, which requires math mode.

\Z is just undefined. My guess is something like:

\documentclass{article}
\usepackage{mathtools}
\usepackage{amssymb}
\usepackage[margin=0.5in]{geometry}
\setlength{\parindent}{0em}
\setlength{\parskip}{1em}

\usepackage{dsfont}
\newcommand*{\Z}{\mathds{Z}}

\begin{document}
Ex. If $x$ is even, then $x^2$ is even.
Proof: Let $y \in \Z\colon x = 2y$
\end{document}

Or using \mathbb:

\documentclass{article}
\usepackage{mathtools}
\usepackage{amssymb}
\usepackage[margin=0.5in]{geometry}
\setlength{\parindent}{0em}
\setlength{\parskip}{1em}

\newcommand*{\Z}{\mathbb{Z}}

\begin{document}
Ex. If $x$ is even, then $x^2$ is even.
Proof: Let $y \in \Z\colon x = 2y$
\end{document}

Also environment equation is not intended for more than one equation. See documentation of package amsmath. It provides many environments (align, gather, ...) for equation systems.

Is this what you wanted?

\documentclass{article}
\usepackage{mathtools}
\usepackage{amssymb}
\usepackage[margin=0.5in]{geometry}
\setlength{\parindent}{0em}
\setlength{\parskip}{1em}

\begin{document}
...
Ex. If $x$ is even, then $x^2$ is even.
Proof: Let $y\in Z : x = 2y$ (we know y exists from the definition of even). Then:
\begin{equation}
\begin{aligned} 
    x^2 &= (2y)^2\\
   \Leftrightarrow x^2 &= 4y^2\\
   \Leftrightarrow x^2 &= 2(2y^2)
   \end{aligned}
 \end{equation}
    $(2y^2)$ is an Integer (because Integers are closed under multiplication), so $x^2$ is Even.
\end{document}