"How can I" export "this executable file outside of the cygwin environment?"
In short: this is impossible. You will need to install the cygwin environment on the target machine and run the programs created in cygwin from the cygwin shell.
Cygwin requires several native .dll files to bind to the underlying Windows OS. They cannot simply be copied to another Windows system without a full cygwin installation.
Here are some more details about this: What is the difference between Cygwin and MinGW?
This is why I prefer to use MinGW to schedule work with Windows systems. Cygwin has the authority and the right to exist when it comes to cross compiling code for various (for example, built-in) purposes running on Windows as a host.
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