OK so I'm doing the quest for Mr. House where you have to go to Cottonwood cove and then Fortification Hill. But the Legion hate me so much they just shoot me on sight, and there are so many of them I die fairly quickly. Do I need them alive to do this quest? Or should I just kill them all?

You can go to Camp Searchlight and ask the NCR Sergent for a mission and when you return he will ask you to kill those Legionaries. Happy days that.

Must I do that to proceed in the main quest? Or can I kill them just with Boone and Rex and still proceed with the main quest?

You can. That's what I did (minus Rex).

Kill 'em allll. But, be prepared for a ridiculous fight, especially at the fort.