Yes. Though their "choice" would probably have been Southeast Asia rather than Ming China or Korea. Whether it was people escaping overseas, people seeking their fortunes overseas, or in the early Edo people being ordered into exile overseas by the Bakufu for being Christians, they seemed to have used the path of Nanban trade, leaving Japan from Sakai/Hirado/Nagasaki, with Manila being the common first stop. One common reason for leaving Japan was afterall religious persecution. Manila's Japan town around 1600 was estimated to have had thousands of Japanese (though not all were exiles, it was a vibrant merchant community). It was certainly enough for the governor to hire 400~500 Japanese to help quell rioting Chinese in 1603, and then for up to 1,500 Japanese themselves to go rioting in 1608. When the Manila authorities complained to the Edo Bakufu in 1608, the Bakufu's response was in summary "not our responsibility, and those who went overseas are basically all criminals so you should just execute them." This is basically the attitude the Edo Bakufu took towards Japanese who lived overseas, and in 1633 the Bakufu forbade all Japanese who lived abroad for over five years from returning to Japan.

Speaking of hiring Japanese, overseas Japanese were a huge source of military manpower in Southeast Asia at the time. Already in the 1580s, the Manila authorities included hiring thousands of Japanese for an invasion plan of China they submitted to the Spanish King (who didn't give his approval). They are quite well attested in the written sources in the wars between the Dutch, English, Spanish, Portuguese. Japanese mercenaries were cheap, and fierce. Jan Pieterszoon Coen, VOC Governor-General of Batavia (that he founded) seemed to have especially liked Japanese mercenaries. However they were also well known for being undisciplined and not following orders, and it seems more than a few times they mutinied. The most famous international incident was probably the the incident at Amboyna in 1623, when 11 Japanese mercenaries of the garrison were caught and under torture confessed to conspiring with English merchants to take the fort, and 9 or 10 of them were executed.

Here are some famous Japanese who went overseas (not necessarily exiled):

• Luson Sukezaemon - As the name suggest, he was a major Nanban merchant operating between Philippines and Japan. For reasons not exactly clear his properties were confiscated by Hideyoshi in 1598, but he escaped overseas, and continued to trade, becoming a major figure in the trade with Cambodia. Interestingly he was specifically named in 1608 Edo's response to Manila as an example of criminal who escaped overseas.
• Yamada Nagamasa - Probably too young to have participated in the Sengoku, but went overseas some time in the early Edo. Later became part of the Japanese mercenary contingent for the King of Ayutthaya (Thailand). The Japanese mercenaries were closely tied to Ayuthaya's military and politics, and Yamada himself was killed and the rest of the Japanese also killed or expelled in 1630 following a coup.
• Kadoya Shichirōbei - Also too young to have participated in the Sengoku. Went overseas on a red-seal ship to Annam (Vietnam) in 1631 only to never return to Japan due to Edo Bakufu's order locking out all those who went overseas, instead becoming a leader of the Japanese community in Annam. Was apparently somehow still able to exchange letters with his family in Matsuoka and continued to send offerings to the famous Ise Grand Shine near by. The tombstone his family erected for him is a Designated Cultural Property of Matsuoka.
• Takayama Ukon - Followed his lord Araki Murashige in his rebellion against Oda Nobunaga, but declared for Nobunaga (kind of) after a bunch of diplomacy (in which the Jesuits were the diplomats), and after Nobunaga's death declared for Hideyoshi. Things were looking up for him until Hideyoshi began his Christian persecution, and he was one of the few Christian lords who gave up his titles rather than his religion. For a time he served under the Maeda clan, but in 1614 was ordered into exile by the Edo Bakufu along with many other Christians. He died soon after arriving in Manila.
• Naitō Tadatoshi - fought and defeated by Nobunaga, and ended up working for Konishi Yukinaga. After Yukinaga was executed following Sekigahara, he ran around a bit, before also ending up with the Maeda clan and then getting exiled with Ukon. He lived in Manilla until his death in 1626.