'It’s like the flip side of being a neat freak' = it is the opposite of being excessively tidy and orderly. 'Flip side' refers to the reverse side of a phonograph record-- there were songs on each side of them. 'Freak' just means someone who is obsessive/preoccupied with a subject or activity.
'Holding on because you’re holding out' -- I cannot figure out how this works into the article, Jenny, but 'holding on' means maintaining one's position or impetus rather than failing or falling behind, while 'holding out' means refusing to consent (to a situation or condition). The sentence seems to suggest that because she will not settle for anything less than a perfectly immaculate house (?), she therefore continues to clean, clean, clean.
Personally, I can't see how she can continue to hold up (= maintain her health)!