Lorentz spinors appear as irreducible representations of the group SL(2,C). Elements of the group are 2x2 matrices with complex entries and unity determinant. A Lorentz spinor is a two component vector $\psi^{A},\chi^{A}\in V_{2}$ with $A=1,2$. The Levi-Civita tensor $\epsilon_{AB}$ is an invariant tensor under SL(2,C). This means that if we have an irrep $\psi^{A}$ of SL(2,C) we can transform it with the Levi-Civita tensor and this will give an equivalent irrep. So, the covariant vectors $\psi_{A}\in \tilde{V}_{2}$ made as $\psi_{A}=\psi^{B}\epsilon_{BA}$ are equivalent to the contravariant vectors $\psi^{A}$; it doesn't matter whether we use $\psi^{A}$ or $\psi_{A}$ because both quantities represent the same physical thing.
The situation is the same as in Minkowski spacetime when we use $x^{\mu}$ or $x_{\mu}=\eta_{\mu\lambda}x^{\lambda}$. Covariant and contravariant vectors in Minkowski spacetime are equivalent irreps of the Lorentz group $O(1,3)$ because the metric $\eta_{\mu\lambda}$ is an invariant tensor.
The only problem with using the Levi-Civita tensor to lower a spinor index is that it is antisymmetric so that it matters which index is summed. I've decided to lower spinor indices as $\psi_{A}=\psi^{B}\epsilon_{BA}$ and so, for consistency, I've got to stick to this convention and not be tempted to use $\psi_{A}=\epsilon_{AB}\psi^{B}$.
Having picked a lowering convention, I'm forced to raise a spinor index with $\psi^{A}=\epsilon^{AB}\psi_{B}$ because then both operations are consistent.
\begin{equation}
\psi_{A}=\psi^{B}\epsilon_{BA}=\epsilon^{BC}\psi_{C}\epsilon_{BA}=\delta^{C}_{A}\psi_{C}=\psi_{A}
\end{equation}
Now we can make a SL(2,C) scalar $\psi_{A}\chi^{A}=\chi^{A}\psi_{A}$. The order of the vectors does not matter because the components $\psi^{1},\psi^{2}$ are just complex numbers. The following bit of index gymnastics recovers the property in the second equation in soliton's question.
\begin{equation}
\psi_{A}\chi^{A}=\psi^{B}\epsilon_{BA}\chi^{A}=-\psi^{B}\chi^{A}\epsilon_{AB}=-\psi^{A}\chi_{A}
\end{equation}
Everything so far has been classical. When we go over to quantum theory, the spinors are promoted to operators $\psi^{A}\rightarrow \hat{\psi}^{A}$. These operators represent fermions: as operators, they have to obey anticommutation relations, in this case $[\hat{\psi}^{A},\hat{\chi}^{B}]_{+}=0$. So, repeating the last calculation with operators,
\begin{equation}
\hat{\psi}_{A}\hat{\chi}^{A}=\hat{\psi}^{B}\epsilon_{BA}\hat{\chi}^{A}=-\hat{\psi}^{B}\hat{\chi}^{A}\epsilon_{AB}=-\hat{\psi}^{A}\hat{\chi}_{A}=+\hat{\chi}_{A}\hat{\psi}^{A}
\end{equation}
recovers the first equation in soliton's question.
I have to own up to the fact that I've not yet studied supersymmetry, but I think this is what must be going on based on general principles.
Best Answer
The epsilon tensors $\epsilon^{\alpha\beta}$, $\bar{\epsilon}^{\dot{\alpha}\dot{\beta}}$ and the sigma symbols $(\sigma^m)_{\alpha\dot{\beta}}$, $(\bar{\sigma}^m)^{\dot{\alpha}\beta}$ are Grassmann-even. Hence, $$\chi_\gamma \epsilon^{\alpha\beta} = \epsilon^{\alpha\beta}\chi_\gamma \,,\quad \epsilon^{\alpha\beta}\epsilon^{\gamma\delta}=\epsilon^{\gamma\delta}\epsilon^{\alpha\beta} \,,\quad \epsilon^{\alpha\beta}(\sigma^m)_{\gamma\dot\gamma}=(\sigma^m)_{\gamma\dot\gamma}\epsilon^{\alpha\beta} \,,\quad \text{etc.} $$ In general, the "invariant tensors" intertwining between representations are Grassmann-even. The $\mathrm{SL}(2,\mathbb{C})$ matrices $L_\alpha{}^\beta$ and $\bar{R}^{\dot{\alpha}}{}_{\dot{\beta}}$ are also Grassmann-even because they value in $\mathrm{SL}(2,\mathbb{C})$.
Also note that there do exist Grassmann-even spinorsã…¡spinor-helicity variables, twistors, Newman-Penrose dyads, etc. The book "Spinors and Space-time" by Penrose and Rindler (my favorite!) provides an excellent introduction to Grassmann-even spinors and their applications (as mentioned also by @KP99 in the comments). An easier version is Peter O'donnell's "Introduction to 2-spinors in general relativity." For the spinor-helicity stuff, I would recommend chapter 2 of Elvang & Huang's "Scattering Amplitudes in Gauge Theory and Gravity."