Defining my own proof environment?

With LaTeX syntax:

\newenvironment{proof}{\paragraph{Proof:}}{\hfill$\square$}

I assume here that you mean \hbox{} not \box in your code. The \null macro is short for \hbox{}. The \box is a TeX primitive and is more like \usebox.

You can also use the amsthm package which is a really nice and quick way of creating a proof.

Implementation

\documentclass{article}
\usepackage{amsthm}

\begin{document}
\begin{proof}
    Your proof here.
\end{proof}

\end{document}

It will add Proof in italics at the beginning of the text given as argument and a white square (Q.E.D. symbol) at the end of it.

I found this easy solution at 1. You can also look up there how you define your own environments using this package.

A small extension due to the case that the proofs are in the section "Proofs:"

\documentclass[11pt]{amsart}
\newtheorem{pro}{Proposition}[section]
\newtheorem{theo}[pro]{Theorem}

\newenvironment{myproof}[2] {\paragraph{Proof of {#1} {#2} :}}{\hfill$\square$}
\begin{document}
\begin{theo} \label{theo}
Hello
\end{theo}

\begin{myproof}{Theorem}{\ref{theo}}
This is the proof
\end{myproof}
\end{document}