No. This is explicitly stated in the paragraph above Theorem 1 of:

Turner, Edward C, Test words for automorphisms of free groups. Bull. London Math. Soc. 28 (1996), no. 3, 255--263.

The author refers to Proposition 1.

No. This is explicitly stated in the paragraph above Theorem 1 of:

Turner, Edward C, Test words for automorphisms of free groups. Bull. London Math. Soc. 28 (1996), no. 3, 255--263.

The author refers to Proposition 1.

EDIT:

Let's give an explicit example. Let $F=\langle a,b\rangle$ and let $g=a[b,a]=ab^{-1}a^{-1}ba$. Clearly $\langle g\rangle$ is a retract of $F$. But the Whitehead graph of $g$ is two triangles glued along an edge. This is Whitehead-reduced, so $g$ is not part of a free basis for $F$.