Yes, a player can use the 'play a cave card for free' ability to bypass the egg cost associated with placing a card in the third or fourth column.

This is explicitly stated in the rulebook on Page 5:

"If you gain a benefit that allows you to excavate for free, you do not need to pay the coin to excavate or the eggs associated with placing a cave card into that column."

This indicates that when using an ability that allows you to play a cave card for free, both the coin cost and the egg cost for the third and fourth columns are waived.

Summary

• The 'play a cave card for free' ability bypasses both the coin and egg costs for placing a cave card in the third or fourth column.
• This rule is found on Page 5 of the rulebook.

Source: Wyrmspan Rulebook (pp. 11, 13, 5, 6)