This problem is 2D in screen coordinates. I have two boxes in screen coordinates, one may assume to be axis aligned but the other box can be any orientation. I want to find the pixels that form their intersection. What is the best algorithm?

SAT

And since it's a bounding box, it's gonna be very fast as you can optimize your check to just 2 axes (parallel line trick).

http://imortisoft.no-ip.org/b2b/?Issue_%231:2D_Convex_Polygon_Collision_using_SAT

If you want to find an intersection, a line to line method should do the trick.