I know I can't pick up NCR quests anymore but will I still be able to recruit all the companions for the achievement?

Well you should but be abit of a challenage to get Rose Cassidy. Make sure you save enough NCR armor. You might get her.
You might have issues completing Boone's Quest at bitter springs.

Joey

Several people have reported simply attempting to recruit them is enough for the achievement even if they shoot you down.

At the time when I first encountered Boone, I wasn't branded a terrorist (Dont Tread on the Bear quest), but I was vilified by the NCR and liked or idolized by Legion.

Boone wouldn't talk to me, so I had to kill Ranger Andy and put on his armor as a disguise. The disguised worked, it changed Boone's attitude and he joined my team.

I tried recruiting Arcade Gannon many times and it was not possible. He was much more difficult than Boone and all the other companions. I didn't get the achievement.