There's no way to access the raw data that were used to create the boxplots unless those values were intentionally saved somewhere in the figure (ie. UserData property) by whoever developed the code that produced those figures.
However, you can access the meta-data used to create the line objects on the plot such as the coordinates of the rectangle, the whisker coordinates, outlier coordinates, etc. You could use those values to simulate the raw data but that won't give you the original raw data.
Here's a demo that shows how to access the meta-data.
ylabel('Miles per Gallon (MPG)')
title('Miles per Gallon for All Vehicles')
bpchil is a list of objects that create the boxplot
Line (Lower Adjacent Value)
Line (Upper Adjacent Value)
Note that the 3rd handle is to the "box". You can get the box coordinates by accessing the XData and YData. Do the same to get the coordinates of the other objects.
bh = findobj(gca,'type','line','tag','Box');
0.925 0.925 1.075 1.075 0.925
Alternatively, you can compute the boxplot statistics from the raw data including the median, upper and lower boxplot edges, and the whisker edges. This is demonstrated at the bottom of this answer (see Extracting location of boxs) using boxchart but you can do the same thing with boxplot by using the inputs.