Elizabeth - load the mat file (in your callback function) and then update the Data property of the uitable with this new cell data. For example,
function pushbutton1_Callback(hObject, eventdata, handles)
myData = load('myMatFile.mat');
The above assumes that the pushbutton to load the mat file is named pushbutton1 and that the table to load the data into is named uitable1. When we invoke the load call, myData becomes a structure whose fields are the variables stored in the mat file. So where I access the cell array with the field myCellArray, you would replace this with the true name of the array (named however you saved it as).