In MegaMek itself, you can save an MUL file of your force before, during, or after a game, and then load it into later games as your "company". You won't get any persistence of experience, and some stuff (like managing pilots HP) is not yet supported in MM.
If you use MekHQ, you can do all your company management there and spawn all or part of it into MM battles. I'm not as familiar with that part, though