I originally traveled with Boone and "fired" him so that I could hire Veronica. Now I've been travelling with Veronica for some time and I've initiated her quest, I Could Make You Care. However, I am hoping to complete Boone's quest as well. So my question is...

If I "fire" Veronica right now, will her quest be botched once and for all or will I be able to re-hire her and pick up where we left off?

I remember firing ED-E several times throughout its quest. Just make sure the companion does not die.

Firing the companions wil not botch their quests. I fire mine all the time so I can tailor my companion to the task at hand. It get much easier after you get the suite in the Lucky 38 and can send them back there. But to answer your question: I fire mine all the time and have never had it mess up any of their companion quests.

-Gunny out.