The Haar basis is unconditional in such spaces iff the so-called Boyd indices are non-trivial (i.e, the lower index is strictly greater than 1 and the upper index is finite). The situation is very much like in $L_p$ spaces. See Lindenstrauss-Tzafriri the second volume, Theorem 2.c.6. You can find the relevant background in the same section.

You can't have an unconditional basis for $L_p$ which is orthonormal in $L_2$ unless $p=2$. I don't know if there are such conditional bases for these spaces though.