In Julius Caesar, what are the two things that Brutus says he will never do, even if he loses the war?

Good question. Both answers are in Act 5, Scene 1.

Even by the rule of that philosophy
By which I did blame Cato for the death
Which he did give himself: I know not how,
But I do find it cowardly and vile,
For fear of what might fall, so to prevent
The time of life....

Brutus says that he finds suicide "cowardly and vile", and that its unphilosophical. Ironic, really, given that by the end of this act he will have run on his own sword.

Cassius then asks him whether he's content to be bound and taken back to Rome in disgrace. Brutus also says that he won't do that either, with characteristic arrogance:

No, Cassius, no. Think not, thou noble Roman,
That ever Brutus will go bound to Rome;
He bears too great a mind.

Hope it helps!