Starcraft – nything missable in the HotS campaign

Campaign progress is stored online. Swapping computers (or reinstalling) shouldn't affect that.

Now that I think about it, I'm not sure if you can save a game in the middle of a mission like you could in SC1, and if so, I'm not sure if that would be saved. But completed missions are completed on other computers.

Update: I have noticed some things that just don't get saved. Like if you finish a mission, then just purchase armory upgrades, then quit. It doesn't always save the armory upgrades. When the campaign is continued, it picks back up immediately after finishing the previous level. But this happens even on the same computer, and isn't any worse if you reinstall StarCraft II

Blizzard on Starcraft 2 Save Games: Starcraft 2 Blog

There is one mission like that in the final few missions that give you an option between 2 missions and depending on what you choose it makes the next mission change a bit, but it is still the same mission.

There is also one hidden objective in one of the missions that will give you access to a secret mission that can only be played by completing this objective, this mission however is not crucial to the story.