Absolutely, definitely, preventing page break

Just enclose your text in an unbreakable unit; before it issue a combination of glue and penalties that will fill the page if the unit has to go to the next one.

\documentclass{article}
\usepackage{lipsum}

\newenvironment{absolutelynopagebreak}
  {\par\nobreak\vfil\penalty0\vfilneg
   \vtop\bgroup}
  {\par\xdef\tpd{\the\prevdepth}\egroup
   \prevdepth=\tpd}

\begin{document}
\lipsum[1]

\begin{absolutelynopagebreak}
\lipsum[1-3]\lipsum*[4]
\end{absolutelynopagebreak}

HERE WE RESTART \lipsum[2]
\end{document}

Try changing \lipsum[1-3] into \lipsum[1-2] and the unit will stay in one page.

One solution may be to use \filbreak before the section, which is defined in plain TeX as \vfil\penalty-200\vfilneg. This suggests that the line is a favorable break location. It worked for a similar problem of mine, at least.

You need to not only prevent page breaks (minipage) but also allow the previous page to stretch (most easily by using \raggedbottom)