According to this study you do want to cook your eggs for better bioavailability. As for soft vs hard, I think you should just go with whatever you prefer.
"The true ileal digestibility of cooked and raw egg protein amounted to 90.9 ± 0.8 and 51.3 ± 9.8%, respectively. A significant negative correlation (r = −0.92, P < 0.001) was found between the 13C-recovery in breath and the recovery of exogenous N in the ileal effluents.
In summary, using the 15N-dilution technique we demonstrated that the assimilation of cooked egg protein is efficient, albeit incomplete, and that the true ileal digestibility of egg protein is significantly enhanced by heat-pretreatment."
** That study measures "bioavailability" by looking at nitrogen content in the breath. Consider the following quote from that same study:"These studies, however, do not take into account the intense intraluminal bacterial metabolism, and absorption and secretion of nitrogen taking place in the colon (Moran and Jackson 1990). The results thus obtained are therefore devoid of any physiologic significance. "**