Create an ArrayList of them.

If I were to create an arraylist of zombies, how would I be able to call of every zombie in the arraylist as a collective to run a method, such as Attack, which would have a bunch of if statements? I can only think of calling each element of the arraylist individually. 

Indeed, that would be how you would do it.

ArrayList<Zombie> armyOfZombies; // To initialize properly

for (Zombie z : armyOfZombies) { z.Attack(); }

If I do that, I'm concerned that it will either ruin the speed of the game or the flow of the game, especially when it reaches like a million zombies which could likely happen. 

Whatever the way you do, there is no magic, each of your zombie has to be called to check what it does and update its status. And modern computers generally can handle well such setting: you worry about performance when you see a bottleneck.

I made my arraylist, and I made a display for the zombies. However, they tend to overlap as they chase around the human character. Is there any way to prevent overlaps, but again without having to call on each and every single x and y of all the other zombies? As of right now, they are represented by white squares on a black background, but i will eventually load images onto the game. 

Usual way is to use brute force. Works well with a limited number of items, shows quickly its limits.
Usual alternative: use a Quadtree.

I did this sketch studying the use of a hash-grid, a simple way of improving performance in collision detection. The code is an update of an old Flux´s sketch, and maybe it´s useful for your purposes. It´s very straight-forward to implement, indeed: http://www.openprocessing.org/visuals/?visualID=34290
 It works well with thousands of elements, depending on the size of the grid buckets. Brute force (to check the distance of any element against the rest directly) works well with hundreds, I ´d say (it depends on so many things: the behaviour algorithms, the graphic display, etc., etc....).
Pd.: What are you working on? A zombie remix of "Gauntlet"? (*sigh*... nostalgia... ) :-)