Hi,
I believe this is a perfect task for a photoshop 'script' since an 'action' might not be able to do it.
Reason being Action cannot take decisions like 'if the image is square' do this else do that... Scripts on the other hand can do that easily. Also action cant remember things like location of file etc.
My proposal is to have a photoshop script, which you can trigger from file>scripts.
Once triggered will show up a popup-UI which will ask user for the following
1. 'input folder'...Browse.. to pick
2. 'output folder'... Browse.. to pick
3. 'watermark file'... Browse to pick psd/png file..
4. quality of output jpg, slider from 1(poor quality) - 12(max quality) [not in your requirment but consider it a bonus :) ]
Once submitted, it will take all files from 'input folder' and the process as per your requirements.. (can prompt for non-square file, or just ignore it).. and save output files in output folder...
If that makes sense, and my profile suits you, lets get it rolling...
Before I end, here is some marketing thing :). Kindly have a look into my profile, checkout my completion rate(100%), and what other clients say about me in reviews.
Looking forward
Cheers