There is no historical background or contemporary account of any of the crusading armies wearing crosses on surcoats. Surcoats worn over chain mail did not appear until the end of the twelfth century anyway, and that was towards the end of the third crusade.
Additionally, both Christian and Muslim warriors wore chain mail and would have looked quite similar, particularly as European armies often took elements of Arabic dress and adapted them to their own personal use (still the case in WW1 & WW2).
The popular image of mail and a white surcoat bearing a red cross is a much later concoction and is more to do with St George, whom if he did exist, was most likely of Middle Eastern origin.
Polish fans dressing as the winged hussars that helped defeat the Muslim army at the siege of Vienna in 1683 would perhaps be more appropriate.
