Instead of using eqnarray, use align (or any of the other environments provided by the amsmath package). During the document preparation, using \allowdisplaybreaks[1] in the preamble, allows an individual automatic page break inside displayed equations; a simple example:
\documentclass{article}
\usepackage[paperheight=4cm]{geometry}% just for the example
\usepackage{amsmath}
\allowdisplaybreaks[1]
\begin{document}
\begin{align*}
a &= b \\
&= c \\
&= c \\
&= c \\
&= c \\
&= c \\
&= c \\
&= c
\end{align*}
\end{document}

As egreg mentions in his comment, once the document is finished, is best to suppress \allowdisplaybreaks[1] from the preamble and use \displaybreak inside the corresponding displayed equation and put it in the best place for the break (which may not be the one automatically chosen); this command is best placed immediately before the line change command where it is to take effect:
\documentclass{article}
\usepackage[paperheight=4cm]{geometry}% just for the example
\usepackage{amsmath}
\begin{document}
\begin{align*}
a &= b \\
&= c \\
&= c \\
&= c \displaybreak\\
&= c \\
&= c \\
&= c \\
&= c
\end{align*}
\end{document}
The reasons for not using eqnarray can be found in Avoid eqnarray! by Lars Madsen.
eqnarrayaltogether and usealignand friends fromamsmath; see eqnarray-vs-align – cmhughes Nov 29 '12 at 00:29