There are 2 approaches.
First, the easiest way is with the 'print' function. Call 'print' in the button callback, and it will get the current figure (which is the GUI) and print it.
For more fine-grained control over the printed image, use the following approach:
f = getframe(gcf);
toShow = figure('Visible','off');
warning('off','images:initSize:adjustingMag')
imshow(f.cdata);
warning('on','images:initSize:adjustingMag')
set(toShow,'PaperOrientation','landscape',...
'PaperUnits','normalized','PaperPosition',[0,0,1,1]);
printdlg(toShow);
close(toShow)
This will capture a picture of the GUI using pixel values, and print an associated figure of the GUI.
To change how the GUI appears on the paper (like the size, location, etc), you can modify some of the properties in the 'set' command. More information is below in the "Printing and Exporting" section: