A friend wrote "I like lazy Sundays"

I responded with, "enjoy them well you can"

Someone corrected me saying I should use "while" instead of "well"

To me I would think both words are correct.

Nah, sorry you're wrong this time. It should be while.

"Enjoy them well you can" sounds like something Yoda from Star Wars would say.

In your version, well is an adverb meaning "in a good way". You wanted while, which is a preposition meaning "for the period".

I hope that helps.