I think the answer to this question is no, as according to my JMP 7 Statistics and Graphing guide (in reference to loss functions for nonlinear modeling):
"(The Loss Function) must have non-zero first and second-order derivatives."
The ABS() function, as it is linear, has a zero second derivative. So, it would seem that the algorithm that JMP uses does not work if you want to use an absolute errors approach. However, I wanted to check as absolute errors isn't all that rare to use, so is it possible that there is some way to work around this?