You have chosen a geographic coordinate system, nonprojected. Mapinfo projects your data using the Plate Carree as a default projection. When you ship the data over to the ESRI house, they use a different default and the likely cause of your disappointment.
You can resolve this by choosing to store your data in a projected scheme. If the map area isn't too extensive, you could use UTM (appropriate zone). If the data is of statewide extent or greater (national), then you should use another projection to avoid distortion.
If stored in a named projection and opened in the same projection, then your shapes will have the same aspect regardless of the software used. But if you leave the decision to DEFAULT, then you'll get the distortion you described. Well, apparent distortion. The projections are doing exactly what they are supposed to do.