(I know that this isn't what you want to do, but it's very close so that others might find this question hoping to do the following.)
egreg's answer is what to do if you only want to change how the reference is typeset when it is referred to. If you want to change it everywhere, meaning when the label itself is typeset, then you need to change a macro called \tagform@. As far as I can see, amsmath doesn't include a facility for changing this easily so it needs to be redefined. Also, its definition involves a few commands that clearly "do something special" so its overall form should be preserved.
Its original definition is:
\def\tagform@#1{\maketag@@@{(\ignorespaces#1\unskip\@@italiccorr)}}
so to change it, you would put the following in your preamble:
\makeatletter
\def\tagform@#1{\maketag@@@{Eq.~(\ignorespaces#1\unskip\@@italiccorr)}}
\makeatother
For an explanation of the surrounding commands, see What do \makeatletter and \makeatother do?
The #1 becomes the equation number. Keep everything inside the \maketag@@@ macro, and keep the stuff surrounding the #1 where it is. Thus to change the format of the equation number (as I did above), you change the opening and closing parentheses and nothing else! (Unless you know what you're doing.)
So if you want your equation numbers to be green with purple spots, you might have:
\makeatletter
\def\tagform@#1{\maketag@@@{\GreenWithPurpleSpots{\ignorespaces#1\unskip\@@italiccorr}}}
\makeatother
Where, \GreenWithPurpleSpots is defined in the StrangeAndBizarreColours package[1].
More seriously, if you want square brackets instead of curved, you could put:
\makeatletter
\def\tagform@#1{\maketag@@@{[\ignorespaces#1\unskip\@@italiccorr]}}
\makeatother
Here's an example showing both this reformatting and egreg's redefinition of the \eqref command:
\documentclass{article}
\usepackage{amsmath}
\makeatletter
\def\tagform@#1{\maketag@@@{[\ignorespaces#1\unskip\@@italiccorr]}}
\makeatother
\let\originaleqref=\eqref
\renewcommand{\eqref}{Eq.~\originaleqref}
\begin{document}
\begin{equation}
\label{aneq}
x^2 + y^2 = z^2
\end{equation}
We refer to \eqref{aneq}.
\end{document}
And the output:

[1]: This is a joke, although I fully expect that package to exist on CTAN within about a week of posting.