Answer:
As long as you travel due east (or west), you will be flying along a line of latitude and you will not cross the equator. Nor is this a great circle. A great circle has its center at the center of the earth.
Great Circle.
Of course, if you start at the north or south pole, you won't be able to travel east or west so the question is undefined at those two points.
Edit: added amplifying information after comments added below.
No matter where you are on the earth, north always points to the north pole, and south always points to the south pole. East and west are perpendicular to those directions. If you point yourself due east and start walking, you will stay at the same latitude.
I think the confusion comes about because really, you cannot walk a straight line on the surface of a sphere. If you want to really travel in a straight line, you would travel off the surface. Think if you had a globe and you thumbtacked one end of a rubber band down on the surface, the only true straight lines will go off the surface. If you lay it along the surface, the rubber band curves along the surface, which is not straight.
So this question is about what if you are moving in what seems like a straight line to you from wherever you are on the earth. Let's try this thought experiment:
Imagine you are one step away from the north pole. Also imagine that there is really short pole at the north pole. You put out your left hand and place it on the pole. You then point yourself due east (which, by the way, is very problematic) and start walking. Your hand will stay on the pole as long as you are traveling east. But you can't walk in a "straight line" and keep your hand on the pole. In fact, it won't be long before you are actually travelling very nearly due south. Then of course you will cross the equator. This actually would be a great circle.
So, you see, it depends on what you mean by a straight line. Do you think lines of latitude are straight lines? Then if you follow that line, you are walking a straight line. Do you think great circle routes are straight lines? Then if you follow one you are walking a straight line. Or if you think the only solution is the true straight line which will take you off the planet then you will not cross the equator (not counting for the earth revolving around the sun and moving underneath you after you start).
This amplifies my original answer a lot, so I will add it to my answer.
Explanation:
