After searching and reading a little, apparently Beverly has quite an interesting history; including the pubs. : )
I found the sign that shows a horse pushing a cart interesting myself and the name 'Push Inn' may relate to an event of such eccentricity but sources only mention a time when the only sign on the door read 'PUSH' and the name stuck.
Then again, I recall a time in my childhood where I saw a cart being pushed exactly like that when we visited a bazaar near my grandmother's house. The vendor was selling green chickpeas which can be consumed like 'potato chips' of our time. Apparently the vibrant green branches of the chickpeas looked very appatizing to the horse and it quickened its pace to walk forth.

The vendor found this a more merciful way of making the horse move instead of heving to use a whip...
Well, look what you made me remember from my childhood Sploosher and Greg!
Yildiz
Link to Pub Names:
http://en.wikipedia.org/wiki/Pub_names