No, a player cannot gain an egg as part of the "any resource" benefit when they tuck a dragon card with an egg cost. According to the rulebook:

• Coins and eggs are not considered resources (resources are defined as gold, crystal, meat, and milk).
• The "any resource" icon in the ability description only allows players to gain one of the defined resources, not coins or eggs.

This is explicitly stated in the rulebook on page 8:

"Coins are not a resource (they’re not one of the colors of the 'any resource' icon), so you can’t gain a coin from this ability."

Eggs are similarly excluded from the definition of resources, so they cannot be gained through this ability.

Summary

Players may only gain gold, crystal, meat, or milk as part of the "any resource" benefit when tucking a dragon card. Eggs are not eligible.

Source: Wyrmspan Rulebook (pp. 11, 2, 8, 9)