Yes
Google should comply with Korea's new Internet regulations.
Google may say that this is a freedom of speech issue, but Google hardly has a perfect reputation with freedom of speech. In China, they run a heavily-censored version of their search engine that complies with all the stringent requirements for control of information that the dictatorial Chinese government requires, so in that case they are not embracing freedom of speech. Restricting freedom of speech in one country in compliance with the government and embracing it in opposition to another government is a little hypocritical.
Also, these regulations are not designed to destroy people's lives, but to protect them. Anti-government groups used the Internet a few months ago to spark a rash of violent protests based off of lies and half-truths about mad cow disease. If these people had been easily identifiable, then the protests could have been nipped in the bud. It would be for the good of the nation.
Besides, the Internet itself needs more regulation. The number of spam emails, spyware, viruses, and criminals floating around on the Internet is just shocking. If there was more real-name identification required for the Internet, these things would not exist.
No
Google should not comply with Korea's new Internet regulations.
The fear of censorship is practically censorship itself. The current Korean administration has made its views on controversial bloggers all too plain with the arrest and detainment of the blogger known as Minerva. Now it only wants to make hunting down and arresting controversial bloggers easier for itself with these new regulations. This has already caused the chill of fear to descend on all Korean internet sites.
Google offers a wide range of services beyond YouTube Korea - blogger.com is a Google service, as is Gmail, Google Docs, and Google Sites. All of these services allow users to upload and distribute content to others. If Google were to compromise first with YouTube Korea, it could then be made to compromise further with its other services. It could eventually be required to make Korean citizens, and Korean citizens only, give their real names and national ID numbers, and to turn over any of them to the Korean government whenever asked.
If Google did that, they would lose customers, and lost customers means lost advertising revenue, and that is Google's bread and butter. Google needs to support freedom of expression to find out what everyone is thinking about, so it can give them targeted text ads, to make money.