I just finished mission 2 (the base capturing mission), and it looks like all of my characters switched jobs to soldiers in between the missions, including my all-important medic.
Is there any way to change jobs between missions (on the character select screen), or do I just need to make it to the next base without losing anyone?

Yes. You can switch professions in missions by going into the specific building & selecting change profession from there.
For example, go to the main building to switch job to Engineer, going to the factory to switch to mechanic & going to the Lab to switch to scientist.
It's very important to read the (! or ?) hints during the game as it gradually guide you through the game.
In fact, I only realised how to switch professions during mission 3.

Yes, you can switch professions in the character selection screen, but your choice of professions depends on the mission.
When you select a character, you choose the profession in the lower left corner from the icons. If everybody switched to soldier, you probably have no other choices there though.

Yea
Changing to:
Soldier = Armory
Mechanic = Factory
Scientist/Medic = Lab
Engineer = Depot
There are some other classes that you will meet in the later parts of the game and i will not say wich and i will not spoil where you upgrade to it ^^
Character selection screen:
When you complete a mission you are given a certain amount of experience points for completing certain things and so on. The experience are automaticly placed on the class you currently are and if you "lvl" up that skill the rest goes to the other classes. So take heed that if you turned one of your Soldiers into a temporary Mechanic chose to have the xp sent to Soldering (if your not ofcourse trying to level the lvls out)
Another note: DO NOT let you characters die... They will NEVER return to you. If the portait are black/dark grey they are dead... If their healthbar is at red, use one of your scientists and make him heal the wounded guy.