When the statistical properties of the underlying data-generating process are "normal", i.e., error terms are Gaussian distributed and iid. In this case, the maximum likelihood estimator is equivalent to the least-squares estimator.
In maximum likelihood estimation, we calculate
$$\hat \beta_{ML}: \sum \frac {\partial \ln f(\epsilon_i)}{\partial \beta} = \mathbf 0 \implies \sum \frac {f'(\epsilon_i)}{f(\epsilon_i)}\mathbf x_i = \mathbf 0$$
the last relation taking into account the linearity structure of the regression equation.
In comparison , the OLS estimator satisfies
$$\sum \epsilon_i\mathbf x_i = \mathbf 0$$
In order to obtain identical algebraic expressions for the slope coefficients we need to have a density for the error term such that
$$\frac {f'(\epsilon_i)}{f(\epsilon_i)} = \pm \;c\epsilon_i \implies f'(\epsilon_i)= \pm \;c\epsilon_if(\epsilon_i)$$
These are differential equations of the form $y' = \pm\; xy$ that have solutions
$$\int \frac 1 {y}dy = \pm \int x dx\implies \ln y = \pm\;\frac 12 x^2$$
$$ \implies y = f(\epsilon) = \exp\left \{\pm\;\frac 12 c\epsilon^2\right\}$$
Any function that has this kernel and integrates to unity over an appropriate domain, will make the MLE and OLS for the slope coefficients identical. Namely we are looking for
$$g(x)= A\exp\left \{\pm\;\frac 12 cx^2\right\} : \int_a^b g(x)dx =1$$
Is there such a $g$ that is not the normal density (or the half-normal or the derivative of the error function)?
Certainly. But one more thing one has to consider is the following: if one uses the plus sign in the exponent, and a symmetric support around zero for example, one will get a density that has a unique minimum in the middle, and two local maxima at the boundaries of the support.
Best Answer
As explained here, OLS is just a particular instance of MLE. Here is closely related question, with a derivation of OLS in terms of MLE.
The conditional distribution corresponds to your noise model (for OLS: Gaussian and the same distribution for all inputs). There are other options (t-Student to deal with outliers, or allow the noise distribution to depend on the input)