Do you have to choose a "kanji name" when you become Japanese?

You do not have to have a "kanji name" when you become Japanese. In fact, I know many naturalized people that opted to not choose kanji for their official legal Japanese name. You don't even have to have a name that "sounds Japanese". I also know many people who chose a Japanese kanji name, but continue to use a katakana form of an English name for their "professional" life in Japan (and often they continue to use their English name outside of Japan). However, you can't have a legal Japanese name that is written in:

• Latin letters (ex. the English alphabet *)
• Even initials are forbidden (JR Sakuragi's given name is not literally "JR" even though it's pronounced that way, and C.W. Nichol's (Ｃ・Ｗニコル {Shī·Daburu NIKORU}) legal given name is not "CW".
• The alphabet is allowed, of course, on Japanese passports, and there is considerable flexibility with respect to how you can write it, "Englishnization" it, and add alternate, additional, or original names.
• Cyrillic script (ex. русский алфавит)
• Arabic script (ex. الحُرُوُفْ العَرَبِيَةُ)
• Korean hangul script (ex. 한글)
• Chinese, Korean, or Vietnamese sinograms — hanzi, hanja — that do not exist in the current Japanese database of allowable sinograms for names
• The database is huge — far beyond what Japanese use in daily life — and grows over time, so most traditional glyphs (繁体字 {hantaiji}), even ones that have been replaced by new Japanese glyphs (新字体 {shinjitai}), are available.
• China and Japan, in the middle of the 20th century, simplified their sinograms. Sometimes, they simplified them in the exact same way. Other times, they did not. Simplified Chinese sinograms (簡体字 {kantaiji}) that does not have a Japanese equivalent cannot be used.
• ... or any other non-Japanese writing system

This can't be your name.

Being forced to adapt your foreign script name for a new countries official writing system is no different from other countries. This isn't just for cultural reasons but also for pragmatic reasons: the technology in the country isn't designed to record, collate, morph, and process all the characters of the world. Additionally, the people recording the name haven't been educated as to how to properly record it. You can't write your name in Cyrillic on most data-processed forms outside of countries that use it.

Even things as simple as diacritics (acute marks over vowels, etc.) have to be removed for countries like the United States.

This means that your new Japanese name must be any arbitrary combination of:

• 平仮名 {hiragana} (Japanese syllabet)
• 片仮名 {katakana} (Japanese syllabet)
• 日本漢字 {Nihon kanji} (Japanese sinograms)

You can theoretically even mix it up within one of your two (you may only have one family name and one given name — no middle names, multi-part family names, suffixes, prefixes, or mononyms) names.

For example, 『えイ度』 {"Eido"} would technically be legal — although the wisdom of doing such a thing is dubious. There are cases, especially for women, where this would appear natural. For example: 『えり子』 {"Eriko"} or 『エリ子』 {"Eriko"}.

"After using a katakana name for so many years in Japan, why would you keep it?"

I've had more than a few people ask me this. In their mind, having a カタカナ {katakana} (Japanese italics-like syllabet) name is something that only non-Japanese use because they don't have a 戸籍 {koseki} (Japanese official family unit/name register) and thus can't have a name that "appears" to be Japanese.

Darvish, spokesman for Seiko's flagship

First of all, it's not true that only foreigners have katakana names. It's not rare for some women (depending on the era and what names and styles were popular) to have all ひらがな {hiragana} (Japanese curvy syllabet) or even all カタカナ {katakana} given names.

As the 21st century progresses, we will see more and more native-born Japan-raised Japanese adults, both ordinary and some very famous and successful, with non-traditional family and given names: MLB professional baseball player (ex. ダルビッシュセファットファリード有 {DARUBISSHU SEFATTO FARĪDO Yū} (Sefat Farid Yu Darvish[sefad]) — aka ダルビッシュ有 {DARUBISSHU Yū} (Yu Darvish) whose Iranian father apparently was not concerned about name length.

In my opinion: 仮名 {kana} (Japanese syllabet) is more "Japanese" than 漢字 {kanji}

Palau, the only state except Japan with
Japanese as an official (minor) language

Kanji is Japanese. But it is also Chinese. And used by many other languages. Kana, on the other hand, was invented and used exclusively in Japan. No other language or country officially in the world uses it (except for the Republic of Palau), making it uniquely Japanese. Kanji, on the other hand, is used in many countries. In my case, I was already borrowing my in-laws' family name, which was kanji and a very traditional and common Japanese family name, so I wanted a legal Japanese name that:

• paid homage to my roots (my father and I have the same given birth name), even if I adapted it for a new culture, as well as my new in-law Japanese family.
• hinted (although it's impossible to be sure without asking, and there are a few (I checked) native born non-foreign-origin Japanese with the name Eido — it's rare though — at my foreign origin.
• when overseas in other 漢字園 {kanjien} (sinogram-using locales) such as China, Taiwan, Singapore, and Korea, I wanted a name that used a script they didn't use. They have names that can be written in Chinese sinograms. My name uses script that is uniquely Japanese. And if I'm going to have a Japanese script name, I am going to be proud of that. Alas, unlike some Asian passports, Japanese passports do not list one's legal Japanese name in kanji. The ローマ字 {rōmaji} (transliterated into Latin letters) name in a Japanese passport is very flexible and can even include additional names, so that script is only for overseas use; it is not an official name for legal domestic use inside Japan.
• was practical for daily and business use: easy to write, remember quickly without repetition, and pronounce

Although having a "kanji name" may be considered "cool" to some — and there's nothing wrong with having a 当て字 {ateji} (foreign sounds transliterated to sinograms) name — I do not think of having or needing a kanji name as being "proof" that I am legally Japanese. I am legally Japanese, and I don't need a legal name that "looks Japanese" to prove it.

Were Naturalized Japanese ever FORCED to take "Kanji Names" or "Japanese Sounding Names"?

Confirmation of official naturalization and other records show that no, there was never an official policy that forced people who took Japanese nationality; you can find many documented examples throughout Japanese history, going on the way back to the 明治時代 {Meiji jidai} (Meiji Era) when the naturalization laws were first created, as well as throughout the eighties & nineties of the 20th century when many people immigrated and naturalized during the wealthy bubble era.

Billionaire (American+Japanese) iPhone fanbois

There are rumors on the English internet that billionaire / Softbank CEO naturalized Japanese (of Korean ethnicity) 孫正義 {SON Masayoshi} (Masayoshi SON) refused to naturalize until they let him keep his Korean name (손정의 {SON Jeong-ui}), but this is not confirmed in any statement or interview that has been published.

Many Koreans in Japan have voluntarily chosen Japanese aliases. The Alien Registration System was even designed to accommodate this in that, until very recently, it allowed all non-Japanese to officially register aliases (通称 {tsūshō}). The alias system was never created with the intent of allowing non-Asian foreigners from setting a Japanese name; it was created to allow 在日 {zainichi} (legally Korean ethnically Japanese) to register a Japanese name. 孫正義 {SON Masayoshi}'s family used the Japanese family name 安本 {YASUMOTO} when he was a child.

It's very conceivable that case officers have strongly recommended or dissuaded people from choosing or not choosing one name or another. Names are extremely important to one's success in society, after all. The evidence, however, does not support the claim that people are or have been forced to take Japanese names.