  The squared Bessel process is a 1-dimensional diffusion process related to
the squared norm of a higher dimensional Brownian motion. We study a model of
$n$ non-intersecting squared Bessel paths, with all paths starting at the same
point $a>0$ at time $t=0$ and ending at the same point $b>0$ at time $t=1$. Our
interest lies in the critical regime $ab=1/4$, for which the paths are tangent
to the hard edge at the origin at a critical time $t^*\in (0,1)$. The critical
behavior of the paths for $n\to\infty$ is studied in a scaling limit with time
$t=t^*+O(n^{-1/3})$ and temperature $T=1+O(n^{-2/3})$. This leads to a critical
correlation kernel that is defined via a new Riemann-Hilbert problem of size
$4\times 4$. The Riemann-Hilbert problem gives rise to a new Lax pair
representation for the Hastings-McLeod solution to the inhomogeneous Painlev\'e
II equation $q"(x) = xq(x)+2q^3(x)-\nu,$ where $\nu=\alpha+1/2$ with
$\alpha>-1$ the parameter of the squared Bessel process. These results extend
our recent work with Kuijlaars and Zhang \cite{DKZ} for the homogeneous case
$\nu = 0$.
