In the case $n \geq 3$ it is false. One way to see this is to use the Pogorelov example $$w(x', \,x_n) = |x'|^{2-2/n}f(x_n),$$ which for the appropriate choice of $f$ solves $\det D^2w = 1$ in $\{|x_n| < \rho\}$. This example is $C^{1,1-2/n}$ and is $0$ on the $x_n$-axis. (This example arises from the affine invariance of the Monge-Ampere equation; $w$ is invariant under the scaling $w(x',\,x_n) = \frac{1}{\lambda^{2-2/n}}w(\lambda x', x_n)$, which preserves the Monge-Ampere measure).
Take $\nu = w$ on $\partial B_{\rho}$. By the comparison principle, $u \geq w$ in $B_{\rho}$, and by convexity $u = 0$ on the $x_n$ axis. Thus, $u$ is not strictly convex, and its regularity is at best $C^{1,1-2/n}$. (One can use analogues of the Pogorelov example of the form $|x'| + |x'|^{n/2}g(x_n)$ in the same way to generate Lipschitz singularities for $u$ along a line segment when the boundary data are Lipschitz).
This example is in a sense optimal; Caffarelli showed that if $\nu$ is $C^{1,\beta}$ for $\beta > 1-2/n$ then solutions to $\det D^2u = \epsilon, \quad u|_{\partial \Omega} = \nu$ are strictly convex and smooth in $\Omega$.
In the case $n = 2$, solutions to $\det D^2u = \epsilon$ are locally strictly convex by a classical result of Alexandrov. By taking smooth approximations $\nu_k$ of the boundary data and solving $\det D^2u_k = \epsilon, \quad u_k|\partial \Omega = \nu_k$ one can hope to obtain the desired approximation in the limit.