How can I apply for a re-entry permit with a U.S. greencard?

Just need to put a reason on the I-131 form itself.

If you apply for the re-entry permit, you must be present in the U.S. at the time you file it, and you need to stay until you complete the biometrics. Then you can leave the U.S. and the re-entry permit will mail to your address. When you come back, you must present your re-entry permit, which normally lasts for two years. Normally, when you return to the U.S. within the two-year period, you should have no problem. If you enter after the expiration of the re-entry permit, it depends on how the CPB officer handles it, they may let you come in without any issues or will conduct the secondary inspection. For US citizenship application, you need to be physically present in the U.S. at least 50% of the time of 5 years before you file your N400 to apply for citizenship. The period of your stay outside the U.S. will definitely affect your qualification for approving your N400 application.