You can prefix most mathematical symbols with \not
in order to negate them. The following looks good according to me:
\documentclass{minimal}
\begin{document}
Did you know that $\mathcal{A}\sqsubseteq\mathcal{B}$? However, $\mathcal{A}\not\sqsubseteq\mathcal{C}$. Also, $\mathcal{C}\not\subseteq\mathcal{D}$.
\end{document}

Alternatively, as suggested by @Alan, there are some variants to this (as well as complete symbols) as part of txfonts
/pxfonts
, mathabx
and mnsymbol
packages. See the page 37 of Scott Pakin's Comprehensive LaTeX Symbol List for more details.