As you may have realized already, there's no ordering of overlapping red, blue, and two white boxcharts that will give you the look you want. You'd need white #1 on top of red, blue on top of white #1, white #2 on top of blue, but red on top of white #2, which is a contradiction.
One thing you can do is to reproduce the built-in boxchart appearance using primitive lines and patches. Of course this requires calculating the statistics like boxchart does, but that's not too hard because boxchart is well-documented.
Something like this:
data1 = randn(100,5);
data2 = randn(100,5);
figure
grid on
box_width = 0.5;
% calculate the median and the box and whisker upper and lower limits for data1
box_med1 = median(data1,1);
q = quantile(data1,[0.25, 0.5, 0.75],1);
box_min1 = q(1,:);
box_max1 = q(3,:);
IQR = box_max1-box_min1;
is_outlier1 = data1 > q(3,:)+1.5*IQR | data1 < q(1,:)-1.5*IQR;
data1_temp = data1;
data1_temp(is_outlier1) = NaN;
whisker_min1 = min(data1_temp,[],1);
whisker_max1 = max(data1_temp,[],1);
% calculate the median and the box and whisker upper and lower limits for data2
box_med2 = median(data2,1);
q = quantile(data2,[0.25, 0.5, 0.75],1);
box_min2 = q(1,:);
box_max2 = q(3,:);
IQR = box_max2-box_min2;
is_outlier2 = data2 > q(3,:)+1.5*IQR | data2 < q(1,:)-1.5*IQR;
data1_temp = data2;
data1_temp(is_outlier2) = NaN;
whisker_min2 = min(data1_temp,[],1);
whisker_max2 = max(data1_temp,[],1);
% create the whiskers and box outlines for data1
N1 = size(data1,2);
xd = (1:N1)-box_width*[0.5;0;0;1;1;0;0;0.5;NaN]/2;
yd = [whisker_min1;whisker_min1;box_min1;box_min1;box_max1;box_max1;whisker_max1;whisker_max1;NaN(1,N1)];
line(xd(:),yd(:),'Color','r')
% create the whiskers and box outlines for data2
N2 = size(data2,2);
xd = (1:N2)+box_width*[0.5;0;0;1;1;0;0;0.5;NaN]/2;
yd = [whisker_min2;whisker_min2;box_min2;box_min2;box_max2;box_max2;whisker_max2;whisker_max2;NaN(1,N2)];
line(xd(:),yd(:),'Color','b')
% create the box interior for data1
xd = (1:N1)-box_width*[0;0;1;1;0]/2;
yd = [box_min1;box_max1;box_max1;box_min1;box_min1];
patch('XData',xd,'YData',yd,'FaceColor','r','FaceAlpha',0.2,'EdgeColor','none')
% create the box interior for data2
xd = (1:N2)+box_width*[0;0;1;1;0]/2;
yd = [box_min2;box_max2;box_max2;box_min2;box_min2];
patch('XData',xd,'YData',yd,'FaceColor','b','FaceAlpha',0.2,'EdgeColor','none')
% create the median line for data1
xd = (1:N1)-box_width*[0;1;NaN]/2;
yd = [box_med1;box_med1;NaN(1,N1)];
line(xd(:),yd(:),'Color','r')
% create the median line for data2
xd = (1:N2)+box_width*[0;1;NaN]/2;
yd = [box_med2;box_med2;NaN(1,N2)];
line(xd(:),yd(:),'Color','b')
% create the outliers line for data1
xd = 1:N1;
[~,idx] = find(is_outlier1);
xd = xd(idx)-box_width*0.25;
yd = data1(is_outlier1);
line(xd(:),yd(:),'Color','none','Marker','o','MarkerEdgeColor','r')
% create the outliers line for data2
xd = 1:N2;
[~,idx] = find(is_outlier2);
xd = xd(idx)+box_width*0.25;
yd = data2(is_outlier2);
line(xd(:),yd(:),'Color','none','Marker','o','MarkerEdgeColor','b')
For comparison, built-in overlapping boxcharts:
figure
hold on
grid on
b1 = boxchart(data1,'BoxFaceColor','r','BoxEdgeColor','r','WhiskerLineColor','r','MarkerColor','r');
b2 = boxchart(data2,'BoxFaceColor','b','BoxEdgeColor','b','WhiskerLineColor','b','MarkerColor','b');
