In the second sentence, they are pretty well interchangeable. In the first sentence, there is a slightly different shade of meaning, which might be increased by the surrounding context. For example:
Someone must stop General Zod! I will back anyone who tries to fight this evil man.
Here, by using the singular someone and the singular anyone, I'm communicating my expectation that that there's going to be a one-on-one confrontation between the evil man and some singular hero.
Hitler will soon move his forces into France. I will back everyone who tries to fight this evil man.
Here, by referencing forces made of many soldiers and using the collective everyone, I'm emphasizing that this is going to be a many-to-many confrontation between the evil man's armies and a collection of heroic soldiers (possibly from many countries) who will fight back.