Kevin,
The following presumes you refer to Windows, as you mention that you use .exes in school. You will not be able to share the environment with OS X. I am unable to check the correctness of the direction I am pointing you to, but it might prove to be a good start.
How would I approach making a "portable" GNUstep build environment for Windows? I would suggest you first install GNUstep on a Windows desktop where you do have admin privileges, then grab the C:\GNUstep folder and copy it to a stick. Then go to another Windows machine which does not have GNUstep and try running various compiler binaries. They are located in \GNUstep\bin.
You will need to familiarize yourself with use of GCC (the compiler), MinGW (the underlying "distribution" of GCC and other tools that GNUstep under Windows is using) and you'll have to figure out how to compile a program using the command line. Sadly, this is out of scope for this mailing list.
What I have described above is what I think I have done with non-GNUstep MinGW a couple of years ago. I have never done it with GNUstep's distribution of MinGW and I have never used it in combination with GNUstep's build system gnustep-make or any other component of GNUstep.
Try whether what I described above works with C, then try writing a program using Objective-C and GNUstep. If something breaks down, let us know, and someone can perhaps try taking a closer look. Right now, considering your posts are a bit unclear, we can mostly encourage you to explore this yourself until you figure out what are the issues that we can actually help you with.
Thank you for your interest in GNUstep!
Thank you for trying to encourage your teachers to consider using C, Objective-C and GNUstep!