\documentclass[12pt]{article}
\usepackage[T1]{fontenc}
\usepackage[utf8]{inputenc}

\usepackage{varioref}
\usepackage{xr-hyper}
\usepackage[colorlinks]{hyperref}
\usepackage{cleveref}
\newtheorem{thm}{Theorem}
\newtheorem{defn}{Definition}
\newtheorem{lem}{Lemma}
\newtheorem{cor}{Corollary}
\newtheorem{prop}{Propostion}

\usepackage{equation}
\externaldocument[A-]{file1} % <==
\externaldocument[B-]{file2} % <==

\begin{document}

This is a test which doesn't work.
(\ref{E:EULERPOISSON})

(\ref{A-eq:file1:1})
These work.
\ref{A-E:EULERPOISSON}
(\ref{A-E:EULERPOISSON})

(\ref{B-eq:file2:1})


(\ref{B-eq:file2:1})

\end{document}