How hard is unshuffling a string?
Michael Soltys and I have succeeded in proving that the problem of determining whether a string can be written as a square shuffle is NP complete. This applies even over a finite alphabet with only $7$ distinct symbols, although our proof is written for an alphabet with $9$ symbols.  This question is still open for smaller alphabets, say with only $2$ symbols. We have not looked at the problem under the restriction that each symbol appears only $6$ times (or, more generally, a constant number of times); so that question is still open.

The proof uses a reduction from $3$-Partition.  It is too long to post here, but a preprint, "Unshuffling a string is $\text{NP}$-hard", is available from our web pages at:

http://www.math.ucsd.edu/~sbuss/ResearchWeb/Shuffle/

and

http://www.cas.mcmaster.ca/~soltys/#Papers.

The paper has been published in the Journal of Computer System Sciences:

http://www.sciencedirect.com/science/article/pii/S002200001300189X