This seems to be a variation of part of the code I provided in this answer
to Beamer: how to change the mini frames from circles to rectangles. Here's is the required code (change the color specifications according to the color schema of your theme, if required):
\documentclass[compress]{beamer}
\usetheme{Singapore}
\usecolortheme{rose}
\usepackage[english]{babel}
\useoutertheme{miniframes}
\definecolor{dark-gray}{gray}{0.10} %color for the navigation squares
\makeatletter
\def\slideentry#1#2#3#4#5#6{%
%section number, subsection number, slide number, first/last frame, page number, part number
\ifnum#6=\c@part\ifnum#2>0\ifnum#3>0%
\ifbeamer@compress%
\advance\beamer@xpos by1\relax%
\else%
\beamer@xpos=#3\relax%
\beamer@ypos=#2\relax%
\fi%
\hbox to 2pt{%
\beamer@tempdim=-\beamer@vboxoffset%
\advance\beamer@tempdim by-\beamer@boxsize%
\multiply\beamer@tempdim by\beamer@ypos%
\advance\beamer@tempdim by -.05cm%
\raise\beamer@tempdim\hbox{%
\beamer@tempdim=\beamer@boxsize%
\multiply\beamer@tempdim by\beamer@xpos%
\advance\beamer@tempdim by -\beamer@boxsize%
\advance\beamer@tempdim by 1pt%
\kern\beamer@tempdim
\global\beamer@section@min@dim\beamer@tempdim
\hbox{\beamer@link(#4){%
\usebeamerfont{mini frame}%
\ifnum\c@section>#1%
\color{dark-gray}%
\else%
\ifnum\c@section=#1%
\ifnum\c@subsection>#2%
\color{dark-gray}%
\else%
\ifnum\c@subsection=#2%
\ifnum\c@subsectionslide>#3%
\color{dark-gray}%
\else%
\color{dark-gray}%
\fi%
\else%
\color{white}%
\fi%
\fi%
\else%
\color{white}%
\fi%
\fi%
\usebeamertemplate{mini frame}%
}}}\hskip-10cm plus 1fil%
}\fi\fi%
\else%
\fakeslideentry{#1}{#2}{#3}{#4}{#5}{#6}%
\fi\ignorespaces
}
\makeatother
\begin{document}
\section{Introduction}
\subsection{Introduction}
\begin{frame}{Frame}
1
\end{frame}
\subsection{Introduction}
\begin{frame}{Frame}
2
\end{frame}
\subsection{Introduction}
\begin{frame}{Frame}
3
\end{frame}
\subsection{Introduction}
\begin{frame}{Frame}
4
\end{frame}
\subsection{Introduction}
\begin{frame}{Frame}
5
\end{frame}
\subsection{Introduction}
\begin{frame}{Frame}
6
\end{frame}
\subsection{Introduction}
\begin{frame}{Frame}
7
\end{frame}
\subsection{Introduction}
\begin{frame}{Frame}
8
\end{frame}
\subsection{Introduction}
\begin{frame}{Frame}
9
\end{frame}
\subsection{Introduction}
\begin{frame}{Frame}
10
\end{frame}
\section{Introduction}
\subsection{Introduction}
\begin{frame}{Frame}
11
\end{frame}
\subsection{Introduction}
\begin{frame}{Frame}
12
\end{frame}
\subsection{Introduction}
\begin{frame}{Frame}
13
\end{frame}
\subsection{Introduction}
\begin{frame}{Frame}
14
\end{frame}
\subsection{Introduction}
\begin{frame}{Frame}
15
\end{frame}
\subsection{Introduction}
\begin{frame}{Frame}
16
\end{frame}
\subsection{Introduction}
\begin{frame}{Frame}
17
\end{frame}
\subsection{Introduction}
\begin{frame}{Frame}
18
\end{frame}
\subsection{Introduction}
\begin{frame}{Frame}
19
\end{frame}
\subsection{Introduction}
\begin{frame}{Frame}
20
\end{frame}
\end{document}
An animation showing the result with the desired scheme for the mini frames:

\documentclass{...}
and ending with\end{document}
. – jub0bs Apr 9 '14 at 14:59