No. Your genetics are fixed and unchanging so 23andme or similar sampling cannot elucidate any damage to your genome. 23andme does not sample the entire genome only around 600,000 or so SNP's which cover what they consider the most active parts of the genome. In other words, the sections that they believe gives them the biggest bang for their buck.
Having said that, 23andme testing is helpful for comparing current symptoms with what is shown in your data. It can give you insights as to 'why' you are responding a certain way. Remember genetics are probabilities not actualities. For instance, it is helpful to know if you are homozygous for MTHFR or other SNP's such as CYP that can effect how you metabolize drugs, which can effect your healing after being floxed. Remember, jut just because it is listed in the genes does not mean you will respond that way.
There was some erroneous information floating around from a supposed credible source in the floxing community that services such as 23andme could detect DNA damage.  Occasionally this rumor will 'make the rounds' so to speak.
I have had my genome sequenced by a couple different companies, 23andme included. I have learned some amazing insights about myself, including why I have some of my symptoms today post floxing. Even though I recommend the floxies eventually patronize a service such as 23andme if possible, I have one word of caution, with these genetic services it is easy to get caught going down a 'rabbit hole', as there is still so much more unknown about the human genome that is known. None the less, they can bring about some fascinating insights, some of which could help you learn how to help yourself heal.
Again, these services are good for learning more about yourself but they cannot identify nor quantify any damage to the nuclear DNA.