I am assuming you want to work with existing USB device drivers across common operating systems, e.g. Windows (several flavours), OS X, and Linux. This would enable a program running on the host PC to open a file, and read the
To send a
1 which is recognised by the host PC, you will need to exchange several message packets. It is not a simple one-bit, or even fixed multi-bit, signal.
Instead the host OS will request enough information from your USB 'button' device to figure out what type of USB device it is, and start the corresponding device driver.
You need to understand how USB works. This USB tutorial USB in a nutshell may help. It describes the protocol in enough detail that it is understandable how the host PC' will interact with the USB device.
To be easy to use, you will probably implement a HID device, which looks like a keyboard to the host PC. The button press could be made to look exactly like a keyboard
1. There are several projects which you can find by searching the web. They all use a microcontroller to manage the quite complex interaction with the host.
One example, which implements USB totally in software using a low-cost microcontroller is V-USB.
There are several other projects, some which do what you are describing, like PJRC's Teensy, and I digispark
Their are lots of MCUs which include a hardware USB interface. Typically they cost more than 1GBP. If you want to go down this path, most of the manufacturers of the USB-enabled MCU provide some sort of library, and their is also the Open Source 'LUFA' USB library.
Summary: the USB protocol is so complex that you will need a microcontroller to implement anything useful. USB is not a simple binary bit pattern.
This V-USB stompbox looks like a close-fit solution, if my other assumptions are correct. It has a single button, emulates a keyboard, and is built on a piece of stripboard (veroboard) with an ATtiny and a few components.
You could use your Arduino to make an AVR ISP programmer. That would enable you to load the firmware onto the ATtiny, so it might be quite a quick, low-cost project.