Not an answer, but I'll just note that there are some special cases that reduce to ODE's.

With $u(x,t) = v(x+at)$, the differential equation becomes

$$ (1+a) v' - a^2 (v'')^2 = v \tag{1}$$

In particular, for $a=0$ we have solutions $u(x,t) = c e^x$ (by symmetry, $u(x,t) = ce^t$ is also a solution), and for
$a=-1$ we have $u(x,t) = -(x-t+c)^4/144$ as well as some solutions that can be expressed in terms of elliptic integrals:
$$x-t = F\left(\sqrt{-u(x,t)}\right)\ \text{where}\ F(z) = \pm \int \dfrac{3 z\; dz}{\sqrt{c \pm 3 z^3}}$$
Standard numerical ODE solvers should work for (1) if you decide on the sign of $v''$, i.e. choose the $\pm$ in

$$ a v'' = \pm \sqrt{(1+a) v' - v}$$

although you are likely to run into trouble if the right side hits $0$.