Will I be eligible for a corporate action if I sell shares on its ex-date?

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Yes, you will be eligible for a corporate action if you sell shares on its ex-date. Shares are settled on T+1. Hence, when you sell shares, they will be credited to the buyer after 1 trading day. In this case, the shares will be in the broker's pool account on the record date and hence as per company records, the corporate action benefit will be credited to the broker. The broker in turn, will credit your account. 

Ex-Date: The date from which a stock starts trading ex-benefit/without the benefits of the corporate action. Investors who buy the share on or after the ex-date are not eligible to receive the benefits.

Record Date: The date on which company records are taken to determine shareholders eligible to receive the benefits of the announced corporate action. Shareholder who hold shares in their demat account on the record date are eligible to receive the benefits.