Let $\mathbb{E}\rightarrow\overline{M}_{g,n}$ be the Hodge bundle. Let us cosider an $F$curve of type $\overline{M}_{1,1}\subseteq\overline{M}_{g,n}$. Is the degree of the restriction of $\mathbb{E}$ on such a curve negative?

No. It's easy to compute that the restriction of $\mathbb E_g$ to $\overline M_{1,1}$ is an extension of the Hodge bundle $\mathbb E_1$ on $\overline M_{1,1}$ and $g1$ copies of the trivial line bundle. The Hodge bundle on $\overline M_{1,1}$ has degree $1/24$. More generally, it is actually true that $\mathbb E$ is nef. 

