How can I use a small Shimano BB on any frame? I have a Shimano BB-ES25 bottom bracket and 45T crankset that was removed from a Giant-CRX City with an eccentric hub (it had Shimano Alfine internal hub gears).

They have literally never been used, and the bike they came out of has since been stolen.

Now I'm planning a custom fixed gear bike build, and hoping to re-use these parts, since I'm on a shoestring budget.

What parts do I need to use this BB in a frame that expects a standard sized BB?

Also, am I correct that, since this is designed for an eccentric hub, I should be able to use any frame for my fixie, instead of being restricted to ones with a horizontal dropout?
The link says that the BB you have is a 68mm shell width, which is standard ISO/British thread.  That's going to fit a lot of bikes.  It is not designed specifically for eccentric bottom brackets.  To use the item you linked to in a single/fixed speed bike you would need horizontal (or at least diagonal) dropouts. 

If that was part of an eccentric BB assembly, you would also have had something sort of like this: 

http://www.jensonusa.com/Misc-Frame-Sup-Bushnell-Eccentric-BB

that it would have gone into, and then that eccentric BB would have gone into an oversized/eccentric bottom bracket shell that was part of the bike frame.  

If you still have that eccentric bottom bracket, then you might be able to go looking for a frame with a same-size eccentric bottom bracket shell that it would fit into.  Such a frame would then have vertical dropouts, because there's no reason to have an eccentric shell and horizontal dropouts in the same frame -- don't need to solve the chain-tensioning problem twice.