Mike,
I'm not sure where you're getting your info, but the tohgr.exe.gz file I downloaded has neither an MZ header nor a PE header anywhere in the file.
Windows executables start with bytes 4D 5A (ASCII "MZ") and have the byte sequence 50 45 00 00 (ASCII "PE" followed by 2 nulls) somewhere in them. The gz archive in question has neither of these.
In fact, the GZIP specification says that gz files start with the fields ID1 and ID2, with values 1F and 8B, respectively. The first two bytes in the tohgr archive are 1F and 8B. I don't think that's a coincidence.
You can find a copy of the spec here: http://www.gzip.org/zlib/rfc-gzip.html#file-format
And what code are you talking about? The binary file? What do you see that makes you think mingw was used, and why is that relevant?
Sorry, but you're not making a lot of sense.