Hold or sell a stock after special dividend announcement result in similar gains?

In theory a dividend payment that has already been announced has an equal impact on the price of the stock. One consideration, however, would be the tax treatment of the dividend vs the tax treatment of the gain, depending on your jurisdiction. In some cases, you may have a better net result receiving the dividend, and in some cases you may have a better net result selling beforehand and having a larger capital gain.

Special dividends can contain capital gains, ordinary income, and/or return of capital. The return of capital component lowers your cost basis and you aren't taxed until you sell your shares. However, the other components are taxable when received.

If you have little to no capital gain in your position, you're in a higher tax bracket, and the taxable component are large, it might make sense to sell before the special dividend in order to avoid the taxable event.

I would not assume that share price will go up will go up by roughly $14 after the announcement because $14 per share is being removed from the company's coffers.