You don't? How about z = 0, 2x + y = 0, for x not equal to 0?

Here z = 0, and therefore x = 0, but y is allowed to be anything except zero. This is all assuming you've done your row reduction correctly. I haven't checked that.

When I take λ = 3 I get

(-2 0 8 ) x = 0

(4 0 12) y = 0

(0 0 4 ) z = 0

and that gives

-2x + 8z = 0

4x + 12z = 0

4z = 0

...

So how am I supposed to find the eigenvectors?