Nair is free out of footstool DJC but the timing is weird. Footstooled opponents can act on frame 29 and Yoshi can act on frame 9. Theoretically this means all of Yoshi's aerials can be done out of DJC before opponents can shield, but the frame window is small for everything except n-air and b-air. Nair is by far the easiest, no explanation needed. B-air is trickier because you can land it with no problems except you'll only get 1-2 hits off unless you hold DJ for a while. D-air and f-air are the same but they have a 4 frame window to input DJ and the aerial, and you'll only get the landing lag hit with d-air if you don't hold DJ for long.
DJC Egg lay is possible with a 4 frame window before they can act. Shielding won't do anything against it so it's a 5 frame window if they buffer a spotdodge (most spotdodges are intangible on frame 2), or 7-8 frame window if they buffer a roll instead. But some characters can punish 1 frame (eg. Bowser up-b, G&W up-b will hit Yoshi and get grabbed on the 1st frame or just hit Yoshi and get out if it's any later, Jigglypuff rest, etc.).
Down-b is only possible on frame perfect footstool->down-b.
So basically n-air and b-air are the only ones worth trying. Yoshi can only act out of footstool some time after he jumps off so find the timing.