How can I get hash browns to bind without using egg?
I follow Alton Brown's recipe from Good Eats for grated (the only way!) hash browns and never have problems with it binding. It's from the episode "Another Man Food Show: Breakfast" if you want to search it out. He talks on the show about why the type of potato matters and recommends a high starch potato like an Idaho or Russet. I use Russets as they're readily available here and never have any issues.

Summary of the cooking process:


10 inch cast iron skillet to medium low. I think I've tried it in a non-stick skillet and while it cooks, you don't get the same browning.
Grate potato with large grater. Squeeze out the water from the grated potato. I use my hands, he uses a tea towel.
Add bacon fat to the skillet. Scatter the potato evenly and cook for 5 minutes without touching it at all. I compress  them down a little bit after I've scattered them. 
Turn down the heat, flip and cook for another 5 minutes. I flip it by sliding it onto a plate, flipping it in the air there, and then sliding it back on. Using a second plate to flip it might be easier.


Notes:


I think the biggest thing is the type of potato. I tried it a few times with some other potato because I had them on hand, and it turned out terrible. I've also tried with previously boiled potatoes, and it was even worse.
Not touching is important too. You want to give them time to stick together.