You actually don't need a supper fast bird. My Argy is a little over 175% movement speed and out ran them easily. Stamina helps though. also it helps to Cary a friend so he or she can grab the egg and shoot the wyvern coming for you.
I got a level 150 lightning egg and I was so excited to hatch the egg...that when I went to put the egg down I accidentally ate it instead. :( ps4 users: R2 ISN'T GONNA DROP THE EGG YOU'RE GONNA EAT IT LIKE ANY OTHER EGG. REMEMBER THIS. LOGIC BEFORE EXCITMENT.