What's the controversy? Biologically there's no such thing as "Jewish DNA". It's just a shorthand for "Human DNA haplotypes (AKA markers) that occur at significant frequencies among populations that identify today as Jewish".
For example, the YDNA haplotype J1, while it occurs at high frequency among Jewish populations, occurs at even higher frequencies among many non-Jewish groups in the Middle East and surrounding areas[1]. It's only somewhat distinctively "Jewish" in areas where Jewish people are a minority like Europe.
Furthermore, the emergence date of this haplotype 17-24k years ago predates the existence of the ethno/religious/cultural identity known as "Jewish" by almost 20,000 years.
Therefore the reverse/opposite of the statement, something like "you can have Jewish DNA and not be Jewish" is either trivially true or nonsensical.