Oops, sorry. I misread your OP. I wouldn't think the components of an EV of a positive-definite matrix would have to be all positive. In fact, any non-zero scalar multiple of an eigenvector is an eigenvector. Hence, if all the components do happen to be positive, just multiply by a negative scalar, and you've got negative components.
Counterexample
Fernando Revilla
Edited: Sorry, I didn' see Ackbeet's post.