They are well-educated.

Should that hyphen be removed and used only when "well-educated" is used as a compound adjective? I know that "well-read" and "well-informed" keep the hyphen either way, but I'm unsure about "well-educated."

No. A hyphen in well educated, well read, well informed, etc., is necessary only when the phrase functions as a modifier.
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Oh, okay, so in a case like with the example that I gave, there should be no hyphen, since there's no word in front of it that it's modifying.

Following your explanation, the phrase "well-educated employees" should be hyphenated.

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