As a soldier doing non-operational work, is there also a prohibition against coming to work on base?

Who said so? Only very specific kinds of activity were prohibited (not all 39 primary categories of labor and their derivatives), and if you are required to come, then you should come.

Is it permitted to work as a medical secretary (scheduling appointments)?

If the employer allows you to take vacation, that is preferable. But if not, then it is possible even on Hol HaMoed. That counts as preventing a loss and meeting holiday needs.

If the employer allows the worker to take vacation but the worker does not want to take vacation on Hol HaMoed, is there a halakhic problem with working on Hol HaMoed in a case where the job does not involve skilled labor? Meaning regular office work and using a computer, which after all anyone does at home on Hol HaMoed.

I understand that in this case there is no prevention of loss, since the employer is willing to let you take vacation.
In my opinion there is a problem. First, almost any labor is skilled labor. Plain typing on a computer maybe not, but writing code or doing bookkeeping is certainly skilled labor.
But beyond that, there is a problem with the very fact of working and hiring yourself out on Hol HaMoed. Even if one is engaged in things he would be allowed to do at home, once it is done for livelihood there is an element of hiring oneself out.
If the work products are for people’s holiday needs, then it is permitted.
I saw a general overview here: https://www.tehillim-center.co.il/article/4854?gclid=CjwKCAjwitShBhA6EiwAq3RqA89xrdC3agvCklqzmFHjLrp5sg3xKC7QNEja1I47ZG3zFnibO3h-kxoC-WsQAvD_BwE