There's a similar question here for circular nodes. You could in principle do this using the \nodeparts command (see section 17.3 of the manual), but I prefer to use a pic (see section 18.2).
Previously, I have defined pic's like this that take many arguments but below I have used \pgfkeys to write a key-value interface, with defaults, because this is much easier to use. The simplest use of the pic is something like
\draw (0,0) pic{splitSq};
to produce:

I have written pic{splitSq={key1=val1, key2=val2,...} to take the following key-value arguments:
size = the size of the rectangle in centimeters (default 1cm)
fill = the fill colour (default gray!10)
top = text in top region
left = text in lower left region
right = text in lower right region
name = tikz name for the node. This also defines names name-T, name-L and name-R for the top, left and right regions, respectively
Consequently, the code:
\begin{tikzpicture}
\draw (6,0) pic{splitSq};
\draw (0,0) pic{splitSq={top=T,left=L,right=R}};
\draw (4,0) pic{splitSq={top=Hi, name=Hi}};
\draw (3,2) pic{splitSq={left=/, fill=blue!10, name=Blue}};
\draw (1,2) pic{splitSq={right=*, fill=green!10, name=Green}};
\draw[->] (Green-R)--(Blue-L);
\draw[->] (Hi)->(Blue);
\end{tikzpicture}
produces:

Here is the full MWE:
\documentclass{minimal}
\usepackage{tikz}
\begin{document}
\pgfkeys{/splitSq/.cd,
size/.initial = 1, % size of square in centimeters
size/.get = \splitSqSize, % store in macro \splitSqSize
size/.store in = \splitSqSize,
fill/.initial=gray!10, % fill colour
top/.initial=, % top text
left/.initial=, % left text
right/.initial=, % right text
name/.initial =, % name
name/.get = \splitSqLabel,
name/.store in = \splitSqLabel
}
\tikzset{
pics/splitSq/.style args = {#1}{% #1=key-value pairs
code = {
\bgroup
\pgfkeys{/splitSq/.cd, #1}% set key-value pairs
\node (\splitSqLabel) at (0,0) [shape=rectangle,
rounded corners,
draw, inner sep=0pt,
fill=\pgfkeysvalueof{/splitSq/fill},
minimum size = \splitSqSize cm] {};
\pgfmathsetmacro\quart{\splitSqSize/4}%
\draw(-2*\quart,0)--++(\splitSqSize,0);% and the lines
\draw(0,0)--++(0,-2*\quart);
\node(\splitSqLabel-L) at (-\quart,-\quart){\pgfkeysvalueof{/splitSq/left}};
\node(\splitSqLabel-R) at (\quart,-\quart) {\pgfkeysvalueof{/splitSq/right}};
\node(\splitSqLabel-T) at (0,\quart) {\pgfkeysvalueof{/splitSq/top}};
\egroup
}
}
}
\begin{tikzpicture}
\draw (0,0) pic{splitSq={top=T,left=L,right=R}};
\draw (4,0) pic{splitSq={top=Hi, name=Hi}};
\draw (3,2) pic{splitSq={left=/, fill=blue!10, name=Blue}};
\draw (1,2) pic{splitSq={right=*, fill=green!10, name=Green}};
\draw[->] (Green-R)--(Blue-L);
\draw[->] (Hi)->(Blue);
\end{tikzpicture}
\end{document}