1: to refuse abruptly
2: to stop short and refuse to proceed
Per Cambridge Dictionary, it means:
to be unwilling to do something or to allow something to happen
So "He did not balk" can be read as "He did not stop" / "He was willing to let things proceed"
You agree balking means being unwilling, right?
So not balking means being willing, right?
He did not balk, therefore he was not unwilling, therefore he was willing, right?
So he was willing to do what Microsoft asked, "but" so was the next guy? What?
No. Balk = stop and return on your steps. He was moving towards ”migrating more than 14000 PCs in public administration to Linux”, Gates and Ballmer tried to make him stop, but he continued on his path.
In this case, willing to proceed with what he was planning to do, that is, moving from Windows to Linux against Microsoft's wishes.
edit: And you're right that this implies "balking at Microsoft's proposal". Personally, I did read the sentence as intended ("they wanted him to balk (= stop his plans), but he did not"), but I can see how the phrasing can be misleading...