+0
Since I hear that you are not supposed to capitalize the first word of quote that precedes the word ‘that’, can you do what the following example demonstrates?

Original sentence: “The Chicken tasted like death.”

Possible quoted revision: Jim said that the “chicken tasted like death.”

Also, what if a proper noun is the beginning of quote that precedes 'that'? Should square brakets be placed around it?
+0
Always preserve the quote. If Jim's full quote was "the chicken tasted like death", then that is how you report it.

Square brackets [ ] are used to make changes to a quote. If the quote contains a pronoun, for example, you can use the name that it refers to in the brackets to avoid confusion.

If the quote is "he said the chicken tasted like death." To clarify, it should be changed to "[Jim] said the chicken tasted like death."

If the quote is "Jim said the chicken tasted like death." then no changes should be made, and you don't need brackets.
Comments  
I beg your pardon, are you mixing up direct quotation (speech marks) and indirect quotation ("that")?

1: Jim said, “The chicken tasted like death.” (direct quotation)
2: Jim said that the chicken tasted like death. (indirect quotation)

Maybe if we change the tense, the difference becomes clearer ...

1: Jim said, “The chicken tastes like death.” (direct quotation)
2: Jim said that the chicken tasted like death. (indirect quotation)
 Vorpar's reply was promoted to an answer.