In J.M.'s Sports and Pastimes, 1675, p. 20 and 24, both the Egg Box (aka Egg Vase) and Globe (aka Ball Box or Vase) are described, respectively. The Egg Box is identical in structure and effect to the Egg and Ball Vases made today in plastic. However, the old Ball Box was not really the same as the Ball Vase we know. The Ball Box, judging from the illustrations given, was a round box without a base or foot; and it came fitted with two shells that nested in the lid. These would both fit over the genuine ball, and the three were different colors. The idea, then, was not that the ball would appear and vanish in the box, but that it changed color twice, before returning to its original color. Therefore, while the two apparatus are clearly related, the effects they were built to achieve were quite different.
Another early explanation of the Egg Vase appears in Jacques Ozanam's Récréations mathématiques et physiques, 1694, p. 101 of the Pieper translation of the 1723 edition. This book was translated in Gibecière, Vol. 6 No. 1, Winter 2011, p. 79-156. In this version, the effects of the Egg Box and Globe have been combined. The box was fitted with three shells, one white, one red and one blue. This allowed both effects. First the egg (made of white marble) is removed from the vase and reappears there. It is apparently removed and swallowed, after which it appears again in the vase. The audience is then asked whether they wish the egg to change to red or blue. It conforms to their choice. To finish, it changes to the third color. So here we have two effects routined: production/vanish with color changes.
Clearly the magicians of the period were varying the effects that might be accomplished with boxes and shells. Judging from these sources, the Egg Vase was the original form, but through combination and evolution (or devolution, if you wish), it became the simple Ball Vase of later centuries.