"Every person" is singular, so the correct way to say it is
"every person needs to consider his/her/their body's needs"
"Their" is acceptable because it is sometimes used as a substitute for "his or her" but it is still singular because of "every person".
"Their bodies' needs" is not always a mistake but it is a mistake in this particular instance. The thing that makes it a mistake is that the transcript says "every person". That means you are talking about just one person (at a time). Since one person has only one body, you cannot say "bodies".
If you said "all people" instead of "every person" the analysis would change.